To Infinity and Back: Logical Limit Laws and Almost Sure Theories

Transcription

1 U.U.D.M. Report 2014:4 To Infinity and Back: Logical Limit Laws and Almost Sure Theories Ove Ahlman Filosofie licentiatavhandling i matematik som framläggs för offentlig granskning den 19 maj 2014, kl 13.15, sal 2005, Ångströmlaboratoriet, Uppsala Department of Mathematics Uppsala University

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3 TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND ALMOST SURE THEORIES OVE AHLMAN This thesis consists of the following papers referred to by their Roman numerals: I O. Ahlman, V. Koponen, Random l-colourable structures with a pregeometry II O. Ahlman, V. Koponen, Limit laws and automorphism groups of random non-rigid structures III O. Ahlman, Countably categorical almost sure theories Introduction History and preliminaries. A cornerstone of mathematical logic is the abstract vocabulary of mathematics. A vocabulary V consists of constant, function and relation symbols where each relation symbol and function symbol has an associated positive integer to it, called its arity. From a vocabulary we may create formulas by adding variables, connectives (e.g. and ) and quantiers (e.g. and ) together with the symbols from the vocabulary. If we want to evaluate some kind of truth of a formula it needs to be a sentence i.e. have no free variables. A quick example from graph theory is the sentence x y(e(x, y) E(y, x)) which states that the edge relation E (part of the vocabulary in graph theory) is symmetric. Dierent parts of mathematical logic do dierent things with languages. The eld of proof theory studies what and how deductions from formulas are possible using certain proof rules. Recursion theory describes what and how fast we may compute functions in models of Peano arithmetic (and things similar to Peano arithmetic). In this thesis we will focus on something in between those elds called model theory, where we discover which sentences are true or false in structures of some (often general) vocabulary and how these structures relate to each other. We will consider vocabularies which only contain relation symbols (called relational vocabularies) making the structures a bit easier to study and avoids certain complications, especially we have the property that any subset of the universe of a structure induces a substructure. A structure with vocabulary V consists of a universe set and, for each ρ N, interpretations of each relation symbol of arity ρ as a set of ordered ρ tuples of elements in the universe. We will denote the structures with calligraphic letters M, N, A, B,... and their universes with the corresponding non-calligraphic letter M, N, A, B,... In order to relate structures 1

4 2 OVE AHLMAN to each other we use embeddings and isomorphisms. An embedding (of relational structures) between V structures A, B is an injective function f : A B such that for each relation symbol R and a 1,..., a r A we have that R is true for the ordered tuple (a 1,..., a r ), noted A = R(a 1,..., a r ), i B = R(f(a 1 ),..., f(a r )). That a function is an embedding between structures will be written f : A B. If f is also a bijection then we say that f is an isomorphism and then A is isomorphic to B, written A = B. For each n N let K n be some set of nite (nite universe) structures over a vocabulary V and let µ n : K n [0, 1] R be a probability measure on K n. The most natural example is to let K n be all V structures with universe {1,..., n} and let µ n be the uniform measure i.e. µ n (N ) = 1 K n for each N K n. In order to study the properties of the structures, we will extend the probability measure µ n so that for a property P (often a sentence in the language) we have that µ n (P) = µ n ({N K n : N satises P}). We may study what happens to the 'really big' structures by regarding lim n µ n (P). This probability does not, for a general K n, even converge. However if a property P is such that lim n µ n (P) = 1 we say that it is an almost sure property. If each sentence ϕ in the language is almost sure or lim n µ n (ϕ) = 0 then we say that K = n=1 K n has a 0 1 law (under the measure µ n ). That a set K has a 0 1 law is hence a proof that it is very well behaved. The study of 0 1 laws for dierent sets K and measures µ n started with Glebskii, Kogan, Liogonkii and Talanov [7] and Fagin [5] who independently proved that for each relational vocabulary V, if K n is the set of all V structures with universe {1,..., n} then K has a 0 1 law under the uniform measure (each structure has the same probability in K n ). It has since then been discovered that many dierent sets have 0 1 laws, spanning from Comptons [3] result about K n being partial orders under the uniform measure to Koponen [11] who proved that an l coloured structure with a pregeometry has a 0 1 law under the dimension conditional measure. In these articles, the method Fagin used, when proving the rst 0 1 law, is adapted to the context and since this method is vital for all three papers we shall take a brief look at Fagins proof. Further examples of 0 1 laws may be found in [1, 2, 4, 8, 9, 10, 13, 14, 15, 16, 17, 18]. The rst thing which is done by Fagin [5] is to nd certain sentences from the language called extension axioms. Each extension axiom speaks about two structures A B such that A + 1 = B and the extension axiom says that for each instance of A we may extend this into the structure B. The proof then continues by proving that these extension axioms are almost surely true in K, which consist of all nite structures of the specied relational vocabulary. A theory (collection of sentences) T is then created by collecting all extension axioms and this theory is shown to be ℵ 0 categorical.

5 TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND ALMOST SURE THEORIES 3 The categoricity follows from a back and forth building of isomorphisms, using that any model of T satises the extension axioms. An ℵ 0 categorical theory, without nite models, is complete and hence any formula is either implied by the theory (thus must have probability 1) or its negation is (thus the formula has probability 0). This theory T, which hence consist of all formulas with probability 1, is called the almost sure theory. For more information about nite model theory the reader is suggested to look at [4]. A pregeometry G = (G, cl) consists of a set G and a closure operator cl : P(G) P(G) acting as a function on the power set of G. The closure operator should satisfy the following axioms for each X, Y G: Reexivity: X cl(x). Monotonicity: X cl(y ) cl(x) cl(y ). Finite character: cl(x) = {cl(x 0 ) : X 0 X and X 0 < ℵ 0 }. Exchange: For a, b G if a cl(x {b}) cl(x) then b cl(x {a}). The most trivial example, called the trivial pregeometry, is the pregeometry where for each X G we have that cl(x) = X. Pregeometries may be viewed as generalizations of vector spaces and we get a direct example by letting G be the vectors and cl be the span operator. In the spirit of vector spaces we say that a set X G is independent if for each a X we have a / cl(x {a}). The dimension of a set X is the size of the largest independent subset Y X. We will however not consider structures with an explicit closure operator, but we may be able to dene it. For a structure M and X M we say that the pregeometry G = (M, cl) is X denable in M if there are formulas θ 0 (x 0 ),..., θ n (x 0,..., x n ),... (possibly with parameters from X) such that for b, a 1,..., a n M we have b cl(a 1,..., a n ) M = θ n (b, a 1,..., a n ). If the pregeometry is denable we may say that M is a pregeometry. About paper I. An old famous problem in mathematics is the study of the colouring of a map or, in mathematical terms, a planar graph. The basic concept is that if we have two vertices adjacent to each other then they need to have dierent colours. For K n being l coloured graphs with universe {1,..., n} Kolaitis et al[10] showed that there is a 0 1 law under the uniform measure. However the concept of colouring is not trivially generalized to vocabularies with higher arities. The key problem is that if we have a relation R(a 1,..., a m ), in some structure, then should all a 1,..., a m have different colours or should only at least two of them have dierent colours? In [11] Koponen discussed a set K n consisting of structures all of which have the same denable pregeometry with dimension n. But in order to give a more general take on l coloured structures, for some xed number l Z +, Koponen studies coloured structures where the underlying pregeometry affects how the colouring is done in the following way: (1) Each element, except those in cl( ), has one of the l colours.

6 4 OVE AHLMAN (2) Any two elements in the same 1 dimensional closure have the same colour. (3) For each relation R(a 1,..., a n ) there are elements b, c cl(a 1,..., a n ) cl( ) such that b and c have dierent colours. Notice that if the underlying pregeometry is the trivial pregeometry then (3) gives us exactly the colouring condition noted above i.e. that at least two elements in a relation needs to have dierent colours. Following this we also want to speak about strongly coloured structures who satisfy the following condition (in addition to (1)-(3)): (4) For each relation R(a 1,..., a n ) and any independent elements b, c cl(a 1,..., a n ) we have dierent colours of b and c. Notice that one important part of being a coloured structure is that there are unary relations P 1,..., P l which represent the colours in the language. Koponen showed that for l coloured structures K n with pregeometries (as described above) and random relations (not involved in the pregeometry) K has a 0 1 law. This generalized the result of Kolaitis et al, which becomes especially clear when the pregeometries are trivial. However, the study of coloured structures becomes much harder when removing the colours from the vocabulary, and just looking at colourable structures. An (strongly) l colourable structure is a structure where we may add l unary relation symbols such that this new structure becomes (strongly) l coloured. Koponen [11] studied the case of K n being just l colourable structures, however she only presented a 0 1 law in the case where the underlying pregeometry is trivial. The big problem when not having a trivial pregeometry is that we have no obvious knowledge about the possible colour of an element, and hence even realizing if a structure is l colourable or not becomes a bit tricky. In paper I we extend the 0 1 law of l colourable structures, to those structures who have a vector space pregeometry (and some closely related variants). This is done by proving that there exists a binary formula ξ(x, y), denable without having explicit colours in the vocabulary, such that almost surely in K, if some structure M = ξ(a, b) then a and b must be assigned the same colour, when colouring the structure M. We then use the formula ξ(x, y) to create extension axioms which suit the class K n and then prove the 0 1 law in the spirit of Fagin's [5] proof. Most of paper I is spent actually creating the formula ξ which is done in surprisingly dierent ways, depending on if we assume the colouring to be strong or not. In the strong case, this is proven through a method inspired by Koponen [12] where we dene the formula by brute force. ξ(x, y) then states that there are so many elements in relation to x and y such that any two elements satisfying ξ(x, y) has to be of the same colour, since we know that (independent) elements in relation to each other have dierent colour, and only l colours exists. The opposite direction, that any two elements with the same colour satisfy ξ, is proven by using the extension properties developed for coloured structures

7 TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND ALMOST SURE THEORIES 5 by Koponen [11] which can induce the structure which ξ(x, y) claims to exist. The normal (non-strong) l colourings on the other hand are proven using a Ramsey theoretic theorem which says that if we have a large enough coloured vector space, then we need to have a large mono-coloured subspace. Using this we create a structure which has as many relations as possible, and yet is still l colourable, in order to x the colouring of the structure. The formula ξ(x, y) then says that x and y are part of the same mono-coloured subspace of this structure, and hence must have the same colour. To show that any elements with the same colour must satisfy ξ(x, y) we again use the extension axioms for coloured structures to nd this large mono-coloured structure in the right place. About paper II. An automorphism on a structure M is an isomorphism from M to M. It is an easy result to show that the set of all automorphisms on M, Aut(M), form a group under compositions. This group acts with a group action on M where one of the more important concepts are the support of an automorphism, which are the elements which are not mapped to themselves. For K n being all V structures with universe {1,..., n} under the uniform measure, Fagin [6] proved that the automorphism group is almost surely the trivial group for structures in K. This has as a consequence that the set K consisting of only nite structures with a trivial automorphism group, has a 0 1 law. One natural follow up to this is to discover what happens if K consists only of structures without trivial automorphism group. These kinds of sets are what we discuss in paper II, where we do a general investigation in the area, discussing what happens when we x the automorphism group or size of the support. In the case of K consisting of all structures we see that the support is almost surely empty, since the automorphism group is almost surely trivial. This does in some way give a hint that the support is almost surely 'small'. In paper II we prove that when comparing the sets of structures K and C where all structures in C has 'a bit larger' support on some automorphism, than the support of any automorphism in K, then Cn K n 0. This is the basis of the paper since it means that whenever we look at structures with large support, or large automorphism group, we may restrict the amount of elements moved by the automorphism. After further work we deduce that we may actually reduce this to having a bounded amount of elements which are actually moved by some automorphism. Using these results we now have a quite a clear picture of how these structures without trivial automorphism group look, one bounded part where the automorphisms can move elements around, and one place which is xed under any automorphisms. This knowledge leads to an estimate of the amount of structures having a certain structure of the elements in the support. Having an estimate of the supports it then becomes easier to compare sets K and C with dierent automorphism group with each other and receive yet another estimate of which of them are more common.

8 6 OVE AHLMAN This paper leads up to a 0 1 law. However it is only possible to deduce it for a very restricted set of structures having a xed structure on the support and the automorphism group being somewhat xed. In the case where we only restrict the support of structures in K or the automorphism groups then we just get a convergence law i.e. the asymptotic probability lim n µ n (ϕ) converges, but not necessarily to 0 or 1. About paper III. In paper III we work in a very general setting using any set of V structures K n under any kind of probability measure µ n, with the only restriction being that the structures in K n grow in size (almost surely) as n grow. Using this general K we study how the almost sure theory of K aects K in the case where K has a 0 1 law under the measure µ n. Especially important for this study are the equivalence relations denable (using formulas from the vocabulary) in the models of the almost sure theories, which we show induce a restriction of which sizes the structures in K may attain. Especially this induces a restriction on the sizes of structures which dene a pregeometry since being independent is an equivalence relation. As mentioned in the preliminaries, Fagin's [5] method for proving a 0 1 law for K consisting of all nite V structure, involves proving that the almost sure theory is ℵ 0 categorical and hence complete. In paper III we do a generalization of this concept and study how an almost sure theory which is ℵ 0 categorical induces properties on the set of structures from which it came from. The result is that the set of structures almost surely satises some kind of extension axioms (not necessarily as easily described as those in [5]), if and only if the almost sure theory is ℵ 0 categorical. So Fagin's method works to prove the 0 1 law of any set K which has an ℵ 0 categorical almost sure theory (and a 0 1 law). Furthermore the sets K with ℵ 0 categorical almost sure theories may be extended in the same way as M eq, getting elements representing equivalence classes. The nice thing about this extension is that the set of extended structures has a 0 1 law if and only if the original set had one, giving us a new way to transform sets of structures in order to nd out if they have a 0 1 law or not. A theory T is strongly minimal if for each model M of T, tuple ā M and formula ϕ(x, ȳ) either ϕ(x, ā) or ϕ(x, ā) is only satised by a nite amount of elements. Paper III ends with a classication of sets K with a 0 1 law whose almost sure theory is both ℵ 0 categorical and strongly minimal. These kinds of theories have quite many nice properties which help in producing the result. The classication comes down to that the sets K who induce these almost sure theories need to have structures where the automorphism group permutes almost all elements. This condition is equivalent with having an ω categorical strongly minimal almost sure theory. Acknowledgments I would like to thank my advisor Vera Koponen, who has put down a lot of work discussing things I have written and who is always positive to my

9 TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND ALMOST SURE THEORIES 7 ideas. My graduate student colleagues at the Department of Mathematics also deserve gratitude, for listening and being part of multiple conversations inside and outside of the area of mathematics. Finally I dedicate this thesis to the shade of my heart, my ancée Karin, for always being encouraging towards me, even when I am not. References [1] S.N. Burris, Number theoretic density and logical limit laws, Mathematical surveys and monographs, Volume 86, American mathematical society (2001). [2] K.J. Compton, A logical approach to asymptotic combinatorics. I. First order properties, advances in mathematics, Volume 65 (1987) [3] K.J. Compton, The computational complexity of asymptotic problems I: partial orders, Inform. and comput. 78 (1988), [4] H-D. Ebbinghaus, J. Flum, Finite model theory, Springer verlag (2000). [5] R. Fagin, Probabilities on nite models, J. Symbolic Logic 41(1976), no. 1, [6] R. Fagin, The number of nite relational structures, Discrete mathematics, Vol. 19 (1977) [7] Y.V. Glebskii, D.I. Kogan, M.I. Liogonkii, V.A. Talanov, Volume and fraction of satis- ability of formulas over the lower predicate calculus, Kibernetyka Vol. 2 (1969) [8] S. Haber, M. Krivelevich, The logic of random regular graphs, Journal of combinatorics, Volume 1 (2010) [9] H.J. Keisler, W.B. Lotfallah, Almost everywhere elimination of probability quantiers, The journal of symbolic logic, Volume 74 (2009) [10] P.G. Kolaitis, H.J. Prömel, B.L. Rothschild, K l+1 -free graphs: asymptotic structure and a 0-1 law, Trans. Amer. Math. Soc. 303 (1987) [11] V. Koponen, Asymptotic probabilities of extension properties and random l-colourable structures, Annals of pure and applied logic, Vol 163 (2012) [12] V. Koponen, A limitlaw of almost l-partite graphs, The journal of symbolic logic, Volume 78 (2013) [13] V. Koponen, Random graphs with bounded maximum degree: asymptotic structure and a logical limit law, Discrete mathematics and theoretical computer science, Volume 14 (2012) [14] J.F. Lynch, Convergence law for random graphs with specied degree sequence, ACM Transactions on computational logic, Volume 6 (2005) [15] J. Spencer, The strange logic of random graphs, Springer (2000). [16] J. Spencer, S. Shelah, Zero-one laws for sparse random graphs, Journal of the american mathematical society, Volume 1 (1988) [17] J. Tyszkiewicz, Probabilities in rst-order logic of a unary function and a binary relation, Random structures & algorithms, Volume 6 (1995) [18] P. Winkler, Random structures and zero-one laws, in N.W. Sauer et al (editors) Finite and innite combinatorics in sets and logic, NATO advanced science institute series, Kluver (1993)

10 Random l-colourable structures with a pregeometry Ove Ahlman and Vera Koponen July 20, 2012 Abstract We study nite l-colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures (with a given underlying pregeometry) by which colours are rst randomly assigned to all 1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satised but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specic colouring of the generating process, has a given property. With this measure we get the following results: 1. A zero-one law. 2. The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. 3. There is a formula ξ(x, y) (not directly speaking about colours) such that, with asymptotic probability 1, the relation there is an l-colouring which assigns the same colour to x and y is dened by ξ(x, y). 4. With asymptotic probability 1, an l-colourable structure has a unique l-colouring (up to permutation of the colours). Keywords: model theory, nite structure, zero-one law, colouring, pregeometry. 1 Introduction We begin with some background. Let l 2 be an integer. Random l-colourable (undirected) graphs were studied by Kolaitis, Prömel and Rothschild in [7] as part of proving a zero-one law for (l + 1)-clique-free graphs. They proved that random l-colorable graphs satises a (labelled) zero-one law, when the uniform probability measure is used. In other words, if C n denotes the set of undirected l-colourable graphs with vertices 1,..., n, then, for every sentence ϕ in a language with only a binary relation symbol (besides the identity symbol), the proportion of graphs in C n which satisfy ϕ approaches either 0 or 1. They also showed that the proportion of graphs in C n which have a unique l-colouring (up to permuting the colours) approaches 1 as n. In [7] its authors also proved the other statements labelled 14 in this paper's abstract, when using the uniform probability measure on C n, although in case of 3 it is not made explicit. This work was preceeded, and probably stimulated, by an article of Erdös, Kleitman and Rothschild [4] in which it was proved that proportion of triangle-free graphs with vertices 1,..., n which are bipartite (2-colourable) approaches 1 as n. One can generalise l-colourings from structures with only binary relations to structures with relations of any arity r 2 by saying that a structure M is l-coloured if the elements of M can be assigned colours from the set of colours {1,..., l} in such that if M = R(a 1,..., a r ) for some relation symbol R, then {a 1,..., a r } contains at least two 1

11 elements with dierent colour. Another way of generalising the notion of l-colouring for graphs, giving the notion of strong l-colouring, is to require that if M = R(a 1,..., a r ) then i j implies that a i and a j have dierent colours. (If the language has only binary relation symbols then there is no dierence between the two notions of l-colouring.) In [8], Koponen proved that, for every nite relational langauge, if C n is the set of l-colourable structures with universe {1,..., n}, then the statements 14 from the abstract hold, for the dimension conditional probability measure, as well as for the uniform probability measure, on C n. The same results hold if we instead consider strongly l- colourable structures. Moreover, the results still hold, for both types of colourings and both probability measures, if we insist that some of the relation symbols are always interpreted as irreexive and symmetric relations. A consequence of the zero-one law for (strongly) l-colourable structures is that if, for some nite relational vocabulary and for each positive n N, K n is a set of structures with universe {1,..., n} containing every l-colourable structure with that universe, and the probability, using either the dimension conditional measure or the uniform measure, that a random member of K n is (strongly) l-colourable approaches 1 as n, then K n has a zero-one law for the corresponding measure; i.e. for every sentence ϕ, the probability that ϕ is true in a random M K n approaches either 0 or 1 as n. In [11], Person and Schacht proved that if F denotes the Fano plane as a 3-hypergraph (so F has seven elements and seven 3-hyperedges such that every pair of distinct elements are contained in a unique 3-hyperedge) and K n is the set of F-free 3-hypergraphs with universe (vertex set) {1,..., n}, then the proportion of hypergraphs in K n which are 2-colourable approaches 1 as n. Since every 2-colourable 3-hypergraph is F-free, it follows the F-free 3-hypergraphs satisfy a zero-one law if we use the uniform probability measure. As another example, Balogh and Mubayi [1] have proved that if H denotes the hypergraph with vertices 1, 2, 3, 4, 5 and 3-hyperedges {1, 2, 3}, {1, 2, 4} and {3, 4, 5} and if K n denotes the set of H-free 3- hypergraphs with universe {1,..., n}, then the proportion of hypergraphs in K n which are strongly 3-colourable approaches 1 as n. Since every strongly 3-colourable 3-hypergraph is H-free it follows that H-free 3-hypergraphs satisfy a zero-one law. In the present article we generalise the work in [8] to the context of (strongly) l- colourable structures with an underlying (combinatorial) pregeometry, also called matroid. Roughly speaking, a structure M with a pregeometry will be called l-colourable if its 1-dimensional subspaces (i.e. closed subsets of M) can be assigned colours from l given colours in such a way that if R is a relation symbol and M = R(a 1,..., a r ), then there are i and j such that the subspaces spanned by a i and by a j, respectively, have dierent colours. A structure M will be called strongly l-colourable if its 1-dimensional subspaces can be assigned colours from l given colours in such a way that if M = R(a 1,..., a r ), then any two distinct 1-dimensional subspaces that are included in the closure of {a 1,..., a r } have dierent colours. The main motivation for this generalisation is to understand how the combinatorics of colourings work out if the elements of a structure are related to each other in a geometrical way, where in particular, the role of cardinality is taken over by dimension. The main examples of pregeometries for which the results of this article apply are vector spaces, projective spaces and ane spaces over some xed nite eld. Another motivation is the fact that pregeometries have played an important role in the study of innite models and one may ask to what extent the notion of pregeometry can be combined with the study of asymptotic properties of nite structures. In [8] a framework for studying asymptotic properties of nite structures with an underlying pregeometry was presented. Here we work within that framework, but since we only consider (strongly) l-colourable structures some notions from [8] become simpler here. 2

12 We now give rough explanations of the notions that will be involved and the main results. Precise denitions are given in Section 2. We x an integer l 2. L pre denotes a rst-order language and for every n N, G n is an L pre -structure such that (G n, cl Gn ) is a pregeometry (Denition 2.1) where the closure operator cl Gn is denable by L pre -formulas (in a sense given by Denition 2.3 and Assumption 2.12). We will consider the property `polynomial k-saturation' (Denition 2.10) of the enumerated set G = {G n : n N}. From Assumption 2.12 it follows that the dimension of G n approaches innity as n tends to innity. The language L rel (from Assumption 2.12) includes L pre and has, in addition, nitely many new relation symbols, all of arity at least 2. By C n we denote the set of all L rel -structures M such that M L pre = G n and M is l-colourable (Denition 2.14). By S n we denote the set of all L rel -structures M such that M L pre = G n and M is strongly l-colourable (Denition 2.14). For every n, δn C denotes the probability measure given by Denition 2.17, which means, roughly speaking, that if X C n, then δn C (X) is the probability that M C n belongs to X if M generated by the following procedure: rst randomly assign l colours to the 1-dimensional subspaces of M, then, for every relation symbol R that belongs to the vocabulary of L rel but not to the vocabulary of L pre, choose an interpretation of R randomly from all possibilities of interpretations R M such that the previous assignment of colours is an l-colouring of the resulting structure, and nally forget the colour assignment, leaving us with an L rel -structure. The probability measure δn S on S n is dened similarly (Denition 2.17). If ϕ is an L rel -sentence then δn C (ϕ) = δn C ( {M Cn : M = ϕ} ) and similarly for δn(ϕ). S Theorem 1.1. Suppose that Assumption 2.12 holds and that G = {G n : n N} is polynomially k-saturated for every k N. Then, for every L rel -sentence ϕ, δ C n (ϕ) approaches either 0 or 1, and δ S n(ϕ) approaches either 0 or 1, as n. If F is a eld and G is the set of vectors of a vector space or of an ane space over F, or if G is the set of lines of a projective space over F, then (G, cl) where cl is the linear closure operator, ane closure operator, or projective closure operator, respectively, forms a pregeometry (see for example [10] or [9]). Theorem 1.2. Suppose that the conditions of Assumption 2.12 hold and that for some nite eld F one of the following three cases holds for every n N: G n is an (a) n-dimensional vector space, or (b) n-dimensional ane space, or (c) n-dimensional projective space, over F, and cl Gn is the linear, ane or projective closure operator on G n, respectively. Moreover, assume that L pre is the generic language L gen from Example 2.4, with the intepretations of symbols given in that example. (i) There is an L rel -formula ξ(x, y) such that the δ C n -probability that the following holds for M C n approaches 1 as n : For all a, b M cl M ( ), M = ξ(a, b) if and only if every l-colouring of M gives a and b the same colour. (ii) lim n δn C ( {M Cn : M has a unique l-colouring} ) = 1. (iii) lim n δn C ( {M Cn : M is not l -colourable if l < l} ) = 1. (iv) The set {ϕ L rel : lim n δn C (ϕ) = 1} forms a countably categorical theory which can be explicitly axiomatised (as in Section 5) by L rel -sentences of the form x ȳψ( x, ȳ) where ψ is quantier-free, mainly in terms of what we call l-colour compatible extension axioms, which involve the formula ξ(x, y) from part (i). (v) The statements (i)(iv) hold if we assume that, for each n N, G n is an n- dimensional vector space over F, cl Gn is the linear closure operator on G n and that 3

13 L pre is the language L F from Example 2.6, with the interpretation of symbols from that example. The assumptions of Theorem 1.2 imply the assumptions of Theorem 1.3, which is explained in Example That is, when dealing with strongly l-colourable structures, the assumptions on the underlying pregeometries can be weaker. By a subspace of a pregeometry we mean a closed set with respect to the given closure operator (Denition 2.3). Theorem 1.3. Suppose that the conditions of Assumption 2.12 hold and that G is polynomially k-saturated for every k N. Also assume that for every n N, every 2- dimensional subspace of G n has at most l dierent 1-dimensional subspaces. (i) There is an L rel -formula ξ(x, y) such that the δ S n-probability that the following holds for M S n approaches 1 as n : For all a, b M cl M ( ), M = ξ(a, b) if and only if every l-colouring of M gives a and b the same colour. (ii) lim n δ S n( {M Sn : M has a unique strong l-colouring} ) = 1. (iii) lim n δ S n( {M Sn : M is not strongly l -colourable if l < l} ) = 1. (iv) Suppose, moreover, that the formulas of L pre which, according to Assumption 2.12, dene the pregeometry G = {G n : n N} are quantier-free. Then the set {ϕ L rel : lim n δ S n(ϕ) = 1} forms a countably categorical theory which can be explicitly axiomatised (as in Section 5) by L rel -sentences of the form x ȳψ( x, ȳ) where ψ is quantierfree, mainly in terms of what we call l-colour compatible extension axioms, which involve the formula ξ(x, y) from part (i). It turns out that Theorem 1.1 follows rather straightforwardly from Theorem 7.32 in [8] when we have proved Lemma 2.21 below. However, Theorem 1.1 in itself does not give information about which sentences have asymptotic probability 1 (or 0), or about properties of the theory consisting of those sentences which have asymptotic probability 1. Neither does it tell us anything about typical properties of large (strongly) l-colourable structures. In order to prove part (i) of Theorems 1.2 and 1.3, which give information of this kind, we treat l-colourable structures and strongly l-colourable structures separately and need to add some assumption(s). The case of strong l-colourings is the easier one and is treated in Section 3; that is, most of the argument leading to part (i) of Theorem 1.3 is carried out in Section 3. The main part of the proof of (i) of Theorem 1.2, dealing with (not necessarily strong) l-colourings, is carried out in Section 4 where we use a theorem from structural Ramsey theory by Graham, Leeb and Rothschild [5]. Once we have established part (i) of Theorems 1.2 and 1.3, which, as said above, is done separately, parts (ii)(iv) (and (v) of Theorem 1.2) can be proved in a uniform way, that is, it is no longer necessary to distinguish between l-colourable structures and strongly l-colourable structures. This is done in Section 5. It is possible to read Section 5 directly after Section 2 and then consider the details of denability of colorings in Sections 3 and 4, which are independent of each other. The theorems above generalise the results of Section 9 of [8] to the situation when a nontrivial pregeometry (subject to certain conditions) is present. In other words, if the closure of a set A is always A (so every set is closed) and we let L pre be the language whose vocabulary contains only the identity symbol `=', and, for every n N, G n is the unique (under these assumtions) L pre -structure with universe {1,..., n+1}, then (i)(iv) of Theorems 1.2 and 1.3 hold by results in Section 9 of [8]. Theorem 1.1 includes this case, without reformulation. 4

14 Remark 1.4. One may want to consider only L rel -structures in which certain relation symbols from the vocabulary of L rel are always interpreted as irreexive and symmetric relations (see beginning of Section 2). Theorems 1.1 and 1.2 hold with exactly the same proofs also in this situation. This claim uses that all results of [8] (see Remark 2.1 of that article) hold whether or not one assumes that certain relation symbols are always interpreted as irreexive and symmetric relations. If a technical assumption is added, explained in Remark 3.7, then Theorem 1.3 also holds in the context when some relation symbols are always interpreted as irreexive and symmetric relations. Remark 1.5. In [8], results corresponding to Theorems , in the case of trivial pregeometries (i.e. when every set is closed), where proved also for the uniform probability measure. The proof used the fact, proved in Section 10 of [8], that, when the pregeometries considered are trivial, then the probability, with the uniform probability measure, that a random (strongly) l-colourable structure with n elements has an l-colouring with relatively even distribution of colours, approaches 1 as n. We believe that the same is true in the context of Theorems 1.2 and 1.3 above, by proofs analogous to those in Section 10 of [8]. But when the underlying pregeometries are no longer assumed to be trivial, then this condition alone seems to be insucient for proving analogoues of Theorems if δn C is replaced by the uniform probability measure on C n and δn S is replaced by the uniform probability measure on S n. In other words, it appears to be a more dicult task to transfer the results of this article to the uniform probability measure (if possible at all) than was the case in [8]. This article ends with a small errata to [8], which makes explicit some assumptions, used implicity in Section 8 of [8], but not stated explicitly in the places in Sections 78 of [8] where they are relevant. 2 Pregeometries and (strongly) l-colourable structures The notation used here is more or less standard; see [3, 9] for example. The formal languages considered are always rst-order and denoted L, often with a subscript. Such L denotes the set of rst-order formulas over some vocabulary, also called signature, consisting of constant-, function- and/or relation symbols. First-order structures are denoted with calligraphic letters A, B,..., M, N,..., and their universes with the corresponding noncalligraphic letters A, B,..., M, N,.... If the vocabulary of a language L has no constant or function symbols, then we allow an L-structure to have an empty universe. Finite sequences/tuples of objects, usually elements from structures or variables, are denoted with ā, x, etc. By ā A we mean that every element of the sequence ā belongs to the set A, and A denotes the cardinality of A. A function f : M N is called an embedding of M into N if, for every constant symbol c, f(c M ) = c N, for every function symbol g and tuple (a 1,..., a r ) M r where r is the arity of g, g N (f(a 1 ),..., f(a r )) = f(g M (a 1,..., a r )), and for every relation symbol R and tuple (a 1,..., a r ) M r where r is the arity of R, M = R(a 1,..., a r ) N = R(f(a 1 ),..., f(a r )). It follows that an isomorphism from M to N is the same as a surjective embedding from M to N. Suppose that L is a language whose vocabulary is included in the vocabulary of L. For any L-structure M, by M L we denote the reduct of M to L. If M is an L-structure and A M, then M A denotes the substructure of M which is generated by the set A, that is, M A is the unique substructure N of M such that A N M and if N M and A N M, then N N. A third meaning of the symbol ` ' with respect to structures is given by Denition Suppose that A is a set, n 2 and R A n an n-ary relation on A. We 5

15 call R irreexive if (a 1,..., a n ) R implies that a i a j whenever i j. We call R symmetric if (a 1,..., a n ) R implies that (π(a 1 ),..., π(a r )) R for every permutation π of {a 1,..., a n }. For any set A, P(A) denotes the power set of A. A usual, we call a formula existential if it has the form where ϕ is quantier free. y 1,..., y m ϕ(x 1,..., x k, y 1,..., y m ) Denition 2.1. We say that (A, cl), with cl : P(A) P(A) is a pregeometry (also called matroid) if it satises the following for all X, Y A: 1. (Reexivity) X cl(x). 2. (Monotonicity) Y cl(x) cl(y ) cl(x). 3. (Exchange property) If a, b A then a cl(x {b}) cl(x) b cl(x {a}). 4. (Finite Character) cl(x) = {cl(x 0 ) : X 0 X and X 0 is nite}. If X, Y A then we say that X is independent from Y if cl(x) cl(y ) = cl( ). From the exchange property it follows that X is indepependent from Y if and only if Y is independent from X (symmetry of independence). We will often write cl(a 1,..., a n ) instead of cl({a 1,..., a n }) and say `a is independent from b' instead of `{a} is independent from {b} over '. We say that a set X is independent if for, each a X, we have that {a} is independent from X {a}. We say that a set X A is closed (in (A, cl)) if cl(x) = X. For X A, the dimension of X is dened as dim(x) = inf{ Y : Y X and X cl(y )}. For more about pregeometries the reader is refered to [9, 10] for example. We will use the following lemma, which has probably been proved somewhere, but for the sake of completeness we give a proof of it here. Lemma 2.2. Let A = (A, cl) be a pregeometry. If {a, v 1,..., v m, w 1,..., w n } A is an independent set then cl(a, v 1,..., v m ) cl(a, w 1,..., w n ) = cl(a) Proof. Suppose that {a, v 1,..., v m, w 1,..., w n } A is an independent set. By reexivity a cl(a, v 1,..., v m ) cl(a, w 1,..., w n ) and so by monotonicity cl(a) cl(a, v 1,..., v m ) cl(a, w 1,..., w n ). For the opposite direction we assume that x cl(a, v 1,..., v m ) cl(a, w 1,..., w n ) and use induction over n to prove that x cl(a). Base case: If n = 0 then cl(a, w 1,..., w n ) = cl(a) so, as x cl(a), we are done. Induction step: Suppose that x cl(a, v 1,..., v m ) cl(a, w 1,..., w n+1 ), so we have two cases to consider: either x ( cl(a, v 1,..., v m ) cl(a, w 1,..., w n+1 ) ) ( cl(a, v 1,..., v m ) cl(a, w 1,..., w n ) ), or x cl(a, v 1,..., v m ) cl(a, w 1,..., w n ). In the rst case we get the consequence that x cl(a, w 1,..., w n+1 ) cl(a, w 1,..., w n ) and hence by the exchange property we get that w n+1 cl(a, w 1,..., w n, x). We already know that x cl(a, v 1,..., v m ) and by also using the assumption that {a, v 1,..., v m, w 1,..., w n } is independent we get that 1 + m + n = dim(a, v 1,..., v m, w 1,..., w n ) = dim(a, v 1,..., v m, w 1,..., w n, x) = 6

16 dim(a, v 1,..., v m, w 1,..., w n, w n+1, x) = dim(a, v 1,..., v m, w 1,..., w n+1 ) = 1 + m + n + 1, so 1 + m + n = 1 + m + n + 1, a contradiction. Hence, x cl(a, v 1,..., v m ) cl(a, w 1,..., w n ), so by the induction hypothesis we get that x cl(a). By induction we conclude that cl(a, v 1,..., v m ) cl(a, w 1,..., w n ) cl(a) holds for all n, which nishes the proof. We will consider rst-order structures M for which there is a closure operator cl on M such that (M, cl) is a pregeometry and, for each n, the relation x n+1 cl(x 1,..., x n ) is denable by a rst-order formula without parameters. More precisely, we have the following denition. Denition 2.3. (i) We say that an L-structure A is a pregeometry if there are L- formulas θ n (x 1,..., x n+1 ), for all n N, such that if the operator cl A : P(A) P(A) is dened by (a) and (b) below, then (A, cl A ) is a pregeometry: (a) For every n N, every sequence b 1,..., b n A and every a A, a cl A (b 1,..., b n ) A = θ n (b 1,..., b n, a). (b) For every B A and every a A, a cl A (B) if and only if a cl A (b 1,..., b n ) for some b 1,..., b n B. (ii) Suppose that A is a pregeometry in the sense of the above denition. Then, for every B A, dim A (B) denotes the dimension of B with respect to the closure operator cl A. In other words, dim A (B) = min{ B : B B and cl A (B ) B}. We sometimes abbreviate dim B (B) with dim(b). A closed subset of A is also called a subspace of A. A substructure B A is called closed if its universe B is closed in (A, cl A ). (iii) Suppose that G is a set of L-structures. We say that G is a pregeometry if there are L-formulas θ n (x 1,..., x n+1 ), for all n N, such that for each A G, (A, cl A ) is a pregeometry if cl A is dened by (a) and (b). It may happen that for an L-structure A there are L-formulas θ n and θ n, for n N, such that the sequence θ n, n N, denes a dierent pregeometry on A (according to Denition 2.3 (i)) than does the sequence θ n, n N. When we use these notions it will, however, be clear that we x a sequence of formulas θ n, n N, and the pregeometry that they dene on each structure from a given set, which will be denoted G. Example 2.4. (Generic example) Every pregeometry (A, cl) can be viewed as a rstorder structure A in the following way. For every n N, let P n be an (n+1)-ary relation symbol and let the vocabulary of L gen be {P n : n N}. For every n N and every (a 1,..., a n+1 ) A n+1, let (a 1,..., a n+1 ) (P n ) A if and only if a n+1 cl(a 1,..., a n ). Then A is a pregeometry in the sense of Denition 2.3 (i) and cl = cl A. It follows that every set of pregeometries G, viewed as L gen -structures is a pregeometry in the sense of Denition 2.3 (iii). Example 2.5. (Trivial pregeometries) If A is a set and cl(b) = B for every B A, then (A, cl) is a pregeometry, called a trivial preometry. Let L be the language with vocabulary, so L can only express whether elements are identical or not. If, for n > 0, θ n (x 1,..., x n+1 ) denotes a formula which expresses that x n+1 is identical to one of x 1,..., x n, and θ 0 (x 1 ) is some formula which can never be satised, then every L -structure is a pregeometry in the sense of Denition 2.3 (i). Moreover, every set G of L -structures is a pregeometry in the sense of Denition 2.3 (iii). 7

17 Example 2.6. (Vector spaces over a nite eld) Let F be a eld. Let L F be the language with vocabulary {0, +} {f : f F }, where 0 is a constant symbol, + a binary function symbol and each f F represents a unary function symbol. Every vector space over F can be viewed as an L F -structure by interpreting 0 as the zero vector, + as vector addition and each f F as scalar multiplication by f. Now add the assumption that F is nite. If, for every n N, θ n (x 1,..., x n+1 ) is an L F -formula that expresses that x n+1 belongs to the linear span of x 1,..., x n, then every F -vector space V, viewed as an L F -structure, is a pregeometry accordning to Denition 2.3 (i). In particular, every set G of vector spaces over a nite eld F, viewed as L F -structures, is a pregeometry according to Denition 2.3 (iii). Denition 2.7. We say that the pregeometry G = {G n : n N} is uniformly bounded if there is a function f : N N such that for every n N and every X G n, clgn (X) ( f dimgn (X) ). Example 2.8. (Vector space pregeometries) Let G = {G n : n N} is a pregeometry. Suppose that, for every n N, (G n, cl G ) is isomorphic (as a pregeometry) with (V n, cl Vn ) where each V n is a vector space of dimension n over a (xed) nite eld F and cl Vn is linear span in V n. Then G = {G n : n N} is uniformly bounded. We get the same conclusion if, instead, each V n is a projective space over F with dimension n, or if each V n is an ane space over F with dimension n. Example 2.9. (Sub-pregeometries of R n ) Let cl n denote the linear closure operator in R n. It is straightforward to verify that whenever X n R n and cl n is dened by cl n(a) = cl n (A) X n for every A X n, then (X n, cl n) is a pregeometry. For every positive integer n choose nite X n R n and, for all n N, let G n = (X n+1, cl n+1). Let L gen be the language from Example 2.4. Then each G n can be viewed as a rst-order structure in the way explained in that example. It follows that G = {G n : n N} is a pregeometry in the sense of Denition 2.3 (iii). Suppose that, in addition, the choice of each X n is made in such a way that for every k > 0 there is m k such that if n > 0 and a 1,..., a k X n, then cl(a 1,..., a k ) X n m k. Then G is uniformly bounded. Denition Let k N. We say that the pregeometry G = {G n : n N} is polynomially k-saturated if there are a sequence of natural numbers (λ n : n N) with lim n λ n = and a polynomial P (x) such that for every n N: (1) λ n G n P (λ n ), and (2) whenever A is a closed substructure of G n and there are G and B A such that A and B are closed substructures of G, G is isomorphic with some member of G and dim G (A) + 1 = dim G (B) k, then there are closed substructures B i M, for i = 1,..., λ n, such that B i B j = A if i j, and each B i is isomorphic with B via an isomorphism that xes A pointwise. Example (i) Let L be the empty language from Example 2.5. It is straightforward to verify that if for every n N, G n is the unique L -structure with universe {1,..., n + 1}, then G is polynomially k-saturated for every k N. (ii) Let F be a nite eld and let L = L gen as in Example 2.4 or L = L F as in Example 2.6. For n N let V n be a vector space over F of dimension n. Each V n gives rise to a pregeometry (V n, cl n ) where cl n is linear span, and each V n can be viewed as an L-structure, call it G n, as in any one of the mentioned examples (depending on whether we take L = L gen or L = L F ). Then the pregeometry G = {G n : n N} is polynomially 8

18 k-saturated for every k N. This is explained in some more detail in [8] and the proofs in Section 3.2 of [2] translate to the present context. (iii) Let F be a nite eld. If G n, for n N, is instead the pregeometry obtained from a projective space over F with dimension n, viewed as an L gen -structure as in Example 2.4, then G = {G n : n N} is polynomially k-saturated for every k N. The same holds if `projective space' is replaced with `ane space'. These facts are proved are proved in a slightly dierent context Section 3.2 of [2], but the proofs there translate straightforwardly to the present context. Assumption We now x some notation and assumptions for the rest of the paper. (1) Let l 2 be an integer, P 1,..., P l unary relation symbols and let V col = {P 1,..., P l }. The symbols P i represent colours. Let V rel be a nite nonempty set of relation symbols all of which have arity at least 2. Let ρ be the maximal arity among the relation symbols in V rel. (2) Let L pre be a language with vocabulary V pre, which is disjoint from both V col and V rel. Suppose that G = {G n : n N} is a set of nite L pre -structures where G n is the universe of G n and G is a pregeometry in the sense of Denition 2.3 (iii). Also, assume that the L pre -formulas θ n (x 1,..., x n+1 ), n N, dene the pregeometry according to Denition 2.3. (3) Let L col be the language with vocabulary V pre V col, let L rel be the language with vocuabulary V pre V rel and let L be the language with vocabulary V pre V col V rel. (4) G is uniformly bounded and, for every n N, if A G n is closed (with respect to cl Gn ) then A is the universe of a substructure of G n (or equivalently, A contains all interpretations of constant symbols and is closed under interpretations of function symbols, if such occur in the language). (5) For every n N, if A is a closed substructure of G n and a 1,..., a n+1 A, then a n+1 cl Gn (a 1,..., a n ) A = θ n (a 1,..., a n+1 ). In other words, the restriction of cl Gn to A is denable in A by the same formulas θ n. (6) For every n N, if A is a closed substructure of G n, then there is m such that A = G m. Also assume that lim n dim(g n ) = and, for every n N, G n cl Gn ( ) = G 0. (7) For every n N, there is a characteristic quantier-free L pre -formula χ Gn (x 1,..., x mn ) of G n, where m n = G n, such that if A is an L pre -structure in which the formulas θ n dene a pregeometry (according to Denition 2.3) and A = χ Gn (a 1,..., a s ) for some enumeration a 1,..., a s of A, then A = G n. Remark (i) If θ n is quantier free for every n N, then (5) holds. Note that in all examples above, it is possible to let θ n be quantier free for every n N, either by using using the generic language L gen from Example 2.4, or by using some of the other languages mentioned in the examples. (ii) Observe that by (5), if A is a closed substructure of G n then the formulas θ n dene a pregeometry (A, cl A ), according to Denition 2.3, and for all X A, cl A (X) = cl Gn (X). By (5)(6), for every k N, there are only nitely many L pre -structures A, up to isomorphism, such that for some n, A G n and dim Gn (A) k. (iii) Condition (7) obviously holds if the vocabulary V pre is nite. But we want to be 9