Constructing Ramsey Graphs from Boolean Function Representations

Transcription

1 Constructing Ramsey Graphs from Boolean Function Representations Parikshit Gopalan College of Computing, Georgia nstitute of Technology, Atlanta, GA 3332, USA. May 1, 2 Abstract Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. hile Erdős probabilistic argument shows the existence of graphs on vertices with no clique or independent set of size, the best explicit constructions achieve a far weaker bound. Constructing Ramsey graphs is closely related to polynomial representations of Boolean functions; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17]. e generalize the above relation by proposing a new framework. e propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in except the origin. e give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-ilson [12], Grolmusz [17] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials. Supported by NSF grant CCR-3B

2 1 ntroduction This paper studies a problem at the intersection of combinatorics and computational complexity. The combinatorial problem is that of explicitly constructing Ramsey graphs. Ramsey s theorem shows that every graph on vertices has either a clique or an independent set of size. n his seminal 17 paper introducing the probabilistic method, Erdős showed that there exist graphs with vertices where [11]. He posed the question of constructing such Ramsey graphs explicitly and offered a prize of "!! for it. This is a central open problem in explicit combinatorial constructions; the best known constructions to date are far from the probabilistic bound. The first breakthrough on this problem was due to Frankl and ilson in 181 [12]; their construction gives $%'& )(+*,-. For over two decades, there was no improvement on this bound despite much effort. However there were other constructions known due to Grolmusz and Alon [17, 2] that achieved exactly the same bound, and also extended to the problem of constructing multi-color Ramsey graphs, which is to. -color the edges of the complete graph so that there is not large monochromatic clique. At first sight, the construction of Grolmusz is quite different from that of Alon and Frankl-ilson, yet it gives exactly the same bound. All three constructions use algebraic techniques, though in different ways. Very recently in 2, the Frankl-ilson bound was beaten by a new construction due to Barak, Rao, Shaltiel and igderson [8] which relies on machinery from pseudorandomness. The complexity problem is to prove tight degree bounds for polynomials computing Boolean functions over /1. A central open problem in circuit complexity is to show lower bounds for ACC, the class of circuits with And, Or and Mod gates. As a first step towards this goal, Barrington, Beigel and Rudich (BBR) studied polynomial representations of Boolean functions modulo composites []. They found surprisingly that such representations are much more powerful over /32 than over /5 when is prime. They showed that the OR function can be represented by symmetric polynomials of degree 78 : ; over /<2. n contrast an = > lower bound is known for the degree of such polynomials over /1. BBR proved a matching = ; lower bound for symmetric polynomials representing the OR function over /?2, and asked if better representations exist using asymmetric polynomials. Tardos and Barrington proved a lower bound of =Ä@CBD5 [22]. This is the best lower bound known for any function, despite much effort [1, 23, 15, 3, 1]. The main open question in this area is whether asymmetric polynomials can give lower degree representations of symmetric functions than symmetric polynomials. A surprising connection between these two problems was discovered by Grolmusz, who used the OR polynomials of BBR to construct Ramsey graphs [17, 18]. As an intermediate step, he constructed a set system of size FEHGJK on elements where all set sizes are! mod but all intersections are non-zero mod L, settling an open problem in extremal set theory. He constructed Ramsey graphs from this set system and showed that lower degree OR representations mod would give better Ramsey graphs. 1.1 Our Results Our work generalizes and extends the connection between OR polynomials and Ramsey graphs. e propose a new definition of an OR representation: a pair of polynomials represent the OR function on variables if the union of their zero sets contains all points in M!NO'P except the origin. e give a simple construction of a Ramsey graph from such representations. This viewpoint based on OR polynomials unifies the constructions of Frankl-ilson, Alon and Grolmusz: they can all be derived from various OR representations of degree 7Q R based on symmetric polynomials. Thus the barrier to better Ramsey constructions through algebraic techniques appears to be the construction of lower degree representations. On one hand, since the best lower bound for any of these representations is only =Ä@CBD there is the possibility of better constructions. On the other hand, we show that further improvements cannot come from representations using symmetric polynomials; we prove an = ; lower bound for such representations. 2

3 1.1.1 Ramsey Graphs from OR Representations: Let ; denote a vector of variables and ; denote a Boolean vector. The following definition of Boolean function representation modulo was introduced by BBR []. / Definition 1 Polynomial Q weakly represents the function mod if for M!NO'P, if then Q Q B. e propose the following definition of an OR representation. / /$% Definition 2 Polynomials and!8 " represent the OR function on and for ) M!NO'P +* Ä!N :! QÄ!N :! Q $8B&3!, B? and!qä!n :! or!8!- B.' $ B(' where ;/' are primes. The degree of the representation is (1.23 5"DF /5"D 7!. variables if One can combine the two polynomials using the Chinese Remainder Theorem (CRT) to get a single polynomial that weakly represents OR mod 8'. However the specific choice of values output by the weak representation is important for our application. The construction of BBR gives a degree 78 OR representation using polynomials over /$ and /:. A simple representation of degree 7Q with ; 8'=< using polynomials over / and /$ can be derived from Alon s construction. This highlights another difference about our definition and weak representations: for Ramsey constructions, we are not restricted to any fixed moduli and ', we are free to choose them in any way, (possibly as functions of ) so that the degree is minimized as a function of. e give a simple Ramsey construction based on OR representations: the vertex set is M!NO'P and we add edge to if >? is in the zero set of Q, denotes the symmetric difference of and. n order to bound and, we use the notion of representations of graphs over spaces of polynomials introduced by Alon [2]. The idea is to assign polynomials to the vertices of so that the polynomials assigned to vertices in a clique are linearly independent. Definition 3 Let 8 7A /B be a graph and C be a set polynomials in variables over field D. A polynomial representation of over D is an assignment of a polynomial FE "@C and a point GE=HD to A where: For JE KE!. f L NHB then JE KMN!. t is easy to see that ;ONP QCQ which is the dimension of the D vector space spanned by polynomials in C. e use the polynomial Q to construct a representation of over /< and!q to construct a representation of over /$. The Frankl-ilson construction can also be viewed in this framework, where we represent over / and over R. However, quoting Alon t seems that this construction does not extend to the case of more than 2 colors [2]. e propose a definition of OR representation which leads to such an extension. Definition Polynomials QÄ!N :! / S and!q!= B&3X " / UT V represent the OR function on and!8ä!n :!,!,8B&?XY variables if and for ) M!NO'P * Ä!N :!!, B?8 or!8!= B&3XY where is prime and Z \[].The degree of the representation is 1V23 5"DF /5"D 7!. 3

4 C / To differentiate the representations of Definitions 2 and, we refer to them as prime representations and primepower representations respectively. The Frankl-ilson construction can be used to show that for <, there exist OR representations of degree 78. The interesting feature of this representation is that it does not use the Chinese Remainder Theorem (CRT). The construction of Ramsey graphs from prime-power representations stays the same; the difference is in the analysis. For this, we introduce polynomial representations of over / S. Definition 5 Let 7A /B be a graph and C a set of polynomials in variables over /. A polynomial representation of over /5 S is an assignment of a polynomial and a point GE= to A s.t.: For JE KE!- B?. f L NHB then JE KMN!- B3. e show that the polynomials assigned to a clique are linearly independent over R so is bounded by the dimension of the R -vector space spanned by C. Like polynomial representations of graphs over R, representations over / S assign linearly independent polynomials over R to vertices in a clique. A crucial difference is that representations over R tensor, those over / S do not. This means that if we have sets of polynomials C and C that represent and over R, then C represents H over R for an appropriate definition of graph product (see [2] for definitions and proofs). This is important for the original application of these representations, which was to bound the Shannon capacity of the graph. However, this property implies that one cannot get low dimensional representations of both and over R, since always has a large clique. But since / S has zero divisors, we lose this tensor product property, so we can simultaneously get low dimensional representations of and over /< S. e could restate this argument from the viewpoint of OR representations. e cannot get low degree prime / representations by since then Q!Q is! at every point in M!NO'P except the origin, and such a polynomial requires degree. But this argument does not extend to prime-power representations because of zero-divisors. All the OR representations above achieve a bound of 7Q ; using symmetric polynomials. Plugging them into the simple construction above gives % & (+*, as opposed to the best bound of % & )(+*,-. However, by massaging the polynomials and working with set intersections as opposed to distances, we can get exactly the constructions of Frankl-ilson, Grolmusz and Alon. Here are some advantages of our unified view of these constructions: t places the constructions of Alon and Frankl-ilson in the context of OR polynomials, and raises the possibility of getting better constructions from low degree representations. The notions of prime-power representations of graphs and Boolean functions arising from the Frankl-ilson construction are of independent interest. t relates the construction of Grolmusz to those of Frankl-ilson and Alon, which look very different at first. Our Ramsey graph construction from the OR polynomial of BBR is simple and direct. n fact it takes some work to show that we get the same graph as Grolmusz. Viewing this construction in terms of set intersections, we derive improved bounds for set systems with restricted intersections modulo prime powers. n this view, all the constructions naturally extend to multicolor Ramsey graphs. To construct. -color Ramsey graphs, we define OR representations involving. polynomials over /- / ; where ' /' are prime powers. Taking powers of the same prime extends the Frankl-ilson construction.

5 1.1.2 Lower Bounds: A natural question is to show tight degree bounds for OR representations. A better upper bound would lead to better Ramsey graphs. Lower bounds are interesting from the complexity-theory viewpoint of understanding polynomial representations over composites. For the OR function, we believe Definition 2 is the right one to use, since it seems to eliminate dependence of the degree on the modulus 8'. Also it places the problem in the context of understanding the zero-sets of low degree polynomials over /. This question has been studied in various other contexts including low degree testing, zero-testing and derandomization (see Section ). Primepower representations are interesting since they do not rely on the CRT. nterestingly, the =Ä@CBD5 lower bound [22] also does not use the CRT, so it applies to prime-power representations too. t is possible that proving bounds for prime-power representations is easier than the prime case. Degree lower bounds extend a line of work in combinatorics aimed at understanding why explicit Ramsey graphs are hard to construct, by showing limitations to various natural techniques. A conjecture of Babai states that one cannot construct good Ramsey graphs based on the sign of a set of real polynomials. There has been considerable progress towards proving this conjecture by Alon et al. [1, ]. Degree lower bounds are weaker since they say the known technique for bounding and does not yield good bounds, as opposed to showing either or is large. But on the other hand, there are no good Ramsey constructions using signs of real polynomials, while OR representations are the best technique known for this problem. Further, the Ramsey graph constructions based on symmetric polynomials result in graphs where the vertex set is M!NO'P and where edges are added between vertices based on the Hamming distance between them. Such graphs possess a high degree of symmetry which is unlikely in a random graph. Our degree lower bound suggests that perhaps such Ramsey graph constructions cannot give better parameters (see Section ). e show a degree = lower bound for OR representations by symmetric polynomials. Thus better representations if they exist must use asymmetric polynomials. A lower bound of = ; is known for symmetric polynomials that weakly represent OR mod L []. Bhatnagar et al. [1] introduced the use of tools from communication complexity for proving degree lower bounds for symmetric polynomials. One might guess that similar arguments should work even for our definition of OR representations, but this is incorrect. n fact those arguments will not suffice even for prime representations. The precise bound they prove, and which holds for all weak representations is 5"D) U5"DF 7!] N 8'. This is good enough when ;/' are small, but if 8' as in Alon s construction, this gives a bound of. One cannot hope for a stronger result since the polynomial of degree weakly represents OR on 8' variables over /U. Our definition restricts the values output by the weak representation, making it possible to show bounds independent of the modulus. But exploiting this difference calls for new techniques, beyond the periodicity based arguments used for weak representations [, 1] Our Techniques: hile lower bounds for the prime and prime-power cases are very different, they have similar high-level structure: an algebraic part where we show that if the zero-set of the polynomial has certain structure, then the polynomial must have high degree, and a combinatorial part where we argue that there is no good partition of hypercube, that any partition results in one of the polynomials having high degree. For the prime-power case, we translate the problem to one about univariate polynomials modulo / S. However over / S it is no longer true that a degree polynomial can have only roots (take for instance); so we need new tools for degree lower bounds. Building on an algorithm for interpolation over / S by the author [1], we define a greedy sequence, which roughly is a sequence that is distributed uniformly among various congruence classes modulo powers of. e show that the longest greedy sequence in the zero-set lower-bounds the degree of a polynomial. Then a combinatorial argument shows that in any partition of integers into and, one of them contains a long greedy sequence. 5

6 / : : For the prime case, we view a symmetric polynomial acting on a -1 vector as a polynomial acting on in the digits of the base expansion of the weight." following Bhatnagar et al.[1]. There it was shown that can be used to bound 5"D) within a factor of ; we introduce a notion of weighted degree of that exactly captures the degree of. The combinatorial part of the proof uses a number theoretic lemma which / / seems of independent interest. t says that if ;/' are primes, X' and and are subsets so that every number in ; lies in mod or in mod ', then one of or has to be large. e present our Ramsey constructions in Section 2. The lower bounds for prime-power representations are in Section 3. n Section 5, we give an alternate construction based on set intersections that results in exactly the constructions of Frankl-ilson, Grolmusz and Alon. e also give improved bounds for set systems with restricted intersections modulo prime powers. e conclude with some open problems in Section. A preliminary version of this paper appears in CCC [13]. 1.2 Preliminaries Let? Ä!N ;:!. Given M!NO'P let. denote its Hamming weight. Given M!NO'P', >? denotes symmetric difference, denotes the bitwise AND and 5 denotes Hamming distance. Let for N ;. Let denote the elementary symmetric polynomial. Every multilinear symmetric polynomial can be written as a linear combination of these polynomials. e will use Lucas Theorem about binomial coefficients modulo which states: Fact 1 Let Then!8 V! B3 For, let the valuation of denoted be the highest power of that divides. Let have the ultrametric inequality $ ];&%' > and ( X N?.! "!. e e say a Ramsey graph 8 7A /B is explicit if there is a deterministic time algorithm to compute the adjacency matrix and very explicit if there is a deterministic Ä@CBD.+-/++ algorithm that computes the adjacency relation. e briefly describe the known explicit constructions of Ramsey graphs in chronological order. < Frankl-ilson [12]: Take prime and. The vertex set consists of all subsets of size <. Two vertices and are adjacent if <$ B3. One can bound the size of and using well-known results from extremal set theory [12, 5].!-8B&QL < Grolmusz [17, 18] : The main step is to construct a set system C on of size EHGJK so that +15+ but +1 7+G!,8B&QL. The vertices of the graph are sets of C and 8 are adjacent if +1 + is odd. One can bound and using results from extremal set theory. < Alon [2]: Take 1' to be nearly equal primes and. The vertex set consists of all subsets of size 8' < <$ B3. To bound and, we construct. Two vertices and are adjacent if +1&:7+8 representations of over / and over /. < Barak [7]: Barak gives a product based construction (discovered independently by Pudlak and Rodl) where we first explicitly search for a good Ramsey graph in a small sample space and then use the Abbot product to get a larger graph. This gives +A +X and <; & F(+*, for any =7>!. A similar product based construction, but with worse parameters is given by Naor [2].

7 / < Barak-Rao-Shaltiel-igderson [8]: n a recent breakthrough, Barak et al. give a construction that achieves '. n fact they solve a more general problem, which is to construct bipartite Ramsey graphs. Their construction is rather intricate and makes significant use of machinery developed for extracting randomness from weak random sources. The first three constructions above are very explicit, the last two are merely explicit. 2 OR Polynomials and Ramsey graphs n this section, we prove the correctness of the construction described in the introduction. hile the graphs obtained are not quite optimal, the construction is simple and best explains the close connections between OR representations and Ramsey graphs. f graph has a representation over a field D as in Definition 3, it is easy to show that ON7 QC where ONP QCQ is the dimension of the D -vector space spanned by C [2]. For representations over / S, we show that? ON7 QC where N7 QC is the dimension of the R -vector space spanned by C. The proof is by a valuation based argument similar to one used by Babai et al. []. Lemma 2 f 7A5/B has a polynomial representation over / S, then 1; N7 QC. PROOF: Let A be a clique. e claim that the polynomials E for are linearly independent over R. Assume for contradiction that E E! Ë By clearing denominators, w.m.a that EH, and by removing common factors w.m.a that does not divide M for some L. Rearranging terms, we have MOJM < E E Ë E M Substituting M, Since M,!,8B&? and is an edge, hence E M M M M < Ë E M E E M M!, we have MOJM KM-P Z <. and L, then LF!- B3. So the RHS is divisible by, which is a contradiction. Construction 1 Graph 8 7A /B from OR polynomials. < Let AQ F M!NO'PO. < f Q >?!, add an edge. Theorem 3 Given a degree OR representation, graph has vertices and. PROOF: Assume that we have a prime representation. e give a polynomial representation of over /. For each vertex M!NO'P, let < 7 if if!

8 Define F to be the polynomial obtained by multi-linearizing Q ; (i.e setting ). Note that for M!NO'P, Hence F1Q -!, B? On the other hand, from our construction, if B then Q H>,!- B3. Hence Q H>!= B&3. This shows that. Thus we get a polynomial representation of over /1. Since the s are all multilinear polynomials of degree at most in variables, they lie in a vector space of dimension Similarly, if is not an edge then Q >!, B&, hence!8 )>!- B('. Using this we construct a representation of over / and bound. For prime-power representations of OR, we can represent and over /< S by the same argument. One can construct explicit Ramsey graphs by plugging in various OR representations described below; all of which give 7Q ; using symmetric polynomials. This gives a bound of % & (+*, for some constant % on the clique size. n fact the constructions below are very explicit, since given vertices M!NO'P, the color of the edge can be computed in time 78 >. / O / Alon [2]: Let ' be primes and let 8'<. Define Q " and!q as $<!Q $< (1) For, since." 8',<, by the CRT.!- B3 or." (!,8B&.'. By Fermat s Theorem, in the former case Q!, B?, in the latter!q!, B.'. Taking /' nearly equal gives degree :. /$ /$: BBR []: Let <. Define Q " and!8 " V as < Q < <!8 (2) < for M!NO'P, Q and!8 in fact have coefficients from / and /$:. For ; Since."." then!q!= B&, <. Lucas theorem implies that if." V!-8B&8 then!- B8, and if!- B. e can choose ) s.t. ' for any >! [1]. Both representations above are prime representations, we now construct prime power representations. For ease of exposition, we restate Definition of prime-power representations in terms of rational polynomials; we omit the simple proof of equivalence. Definition Polynomials!Q and for ) M!NO'P +* Ä!N :! QÄ!N :! Q HR represent the OR function on M!NO'P $ B3!, B& and!qä!n :! or!8!- B? for a prime. The degree of the representation is "D) / 5"DF 7!. 8 $ B3 if

9 Note that in general could be rational. hen we say Q! - B&, we mean is an integer satisfying the condition. However, if and!8!- B3, then Q need not be an integer and vice versa. Frankl-ilson [12]: Take prime and <. Define Q!8 $ R as Q <!8 For a non-zero vector? M!NO'P we have. $ f.!, B?, then GK < hence!q < (3) <. f. = V B then Q!= B&.!, B?. The degree is <. <. Define e construct representations with the prime fixed and varying, analogous to []. Let Q!8 $ R as < Q < <!8 < () The proof of correctness is through Lucas theorem. f.!- B8 then!. f! but."!= B& then!q!. Plugging any of these polynomials into construction 1 gives the following type of graph : Add to B if ),!, B where is either a prime or a prime power close to. For constructing. -color Ramsey graphs, we define OR representations with. polynomials / such that the union of their zero sets if M!NO'P * M FP. e can extend constructions in Equations 1, 2 by taking. distinct primes. To extend the construction of Equation 3, let <. For. define O < N (5) e can similarly extend Equation, we omit the details. 3 Lower Bounds for Prime-power Representations n this section we prove a lower bound for prime-power representations by symmetric polynomials. / S / UT Theorem Let Q " and!q " be symmetric polynomials that represent the OR function on variables. Then 5"DF 5"D) 7! "]. Since a symmetric polynomial on -1 inputs is essentially a polynomial in the weight of the input, we can restate Theorem in terms of integer polynomials. The formal proof is easy and is omitted. hile we could also work with polynomials in / S, the presence of zero divisors would make it messier to use valuations. /- Proposition 5 Let!8 $ be univariate polynomials such that for M P, QÄ! P ; FP or!qä! P!8 P () Then 5"D) 5"DF 7! $].

10 The next two Lemmas ( and 7) develop tools to show degree bounds for such polynomials. Definition 7 A sequence of integers is called a greedy sequence if for all, < < for Let us define " and " < for >. The definition of a greedy sequence can be restated as P P for ). Given any set, we can order it elements greedily as follows to get a greedy sequence: we choose arbitrarily; having chosen ; we choose 2 to be the element that minimizes P. Lemma Let /- be a greedy sequence. Let Q be such that Q P Q P for N ; < Then 5"D "] +<. PROOF: The proof is by induction on. e will show the converse, namely that if 5"DF 1;+<. Q P P ] &%' The base case V is trivial, in this case is constant so it is clear that PJ P. Assume the property holds for greedy sequences of Given a polynomial Q of degree <, since is a monic polynomial of degree <, we write Q!Q %, where!8 is a polynomial of degree +<. Substituting, Q!Q % hence by the ultrametric inequality Q PH] &%' M!8 P % PP (7) To lower bound!q P, note that the sequence of length < obtained by deleting is also greedy. Hence applying the inductive hypothesis to!q, we get!q PV Q P (8)!8 P ] &%' &%' The last equality follows since! for N8 <, hence!8 % P. Using the greedy property of the sequence ;, % P ] % P % Q <!8 Hence we have Since % PH] &%')M Q is a greedy sequence and! has degree +<!8!Q PV P ] &%' 1 P &%'!Q Q, we get by induction that Q. e now lower bound PP () P (1)

11 P P Combining Equations 7, 8,, 1 gives the desired result. An example of a greedy sequence is when are consecutive integers. Thus while a degree polynomial over / S can have several zeroes, the lemma implies that it can have at most consecutive zeroes. The intuition for this Lemma is from an algorithm for polynomial interpolation over / S by the author [1]. Given a set, and values F for of some polynomial in / S, the algorithm will output the smallest degree polynomial that fits the data, provided it sees the elements of in the above greedy order. f the polynomial is! on every element but the last, the algorithm is forced to output a polynomial of degree +<. Next we define the notion of a greedy array which we use to construct long greedy sequences. Given a. -dimensional matrix of dimension, we use to denote N N. Definition 8 A. -dimensional matrix of distinct integers is called a greedy array if < &X1&%')MUZ+ N P (11) e define an ordering of the array indices, which is essentially the reverse lexicographic (revlex) ordering. Definition Given a. -dimensional integer vectors and, let 1.23)MUZ + N P. Then if N. Note that for a greedy array, the valuation should equal the smallest index where and differ. However to order elements, we look at the largest index where they differ. For example, consider the array where N ;N J N N = /N and! Thus the array contains N M!N A < 'P with numbers indexed by their base-p expansion. Since N < depends on the smallest digit where N and, this is a greedy array. The ordering defined above is the usual ordering of integers, it depends on the largest digit where the expansions differ. Lemma 7 Ordering elements of a greedy array gives a greedy sequence. PROOF: e want to show that for 8< X8< & (12) For! ;Z. <, we define the set M + N N ; N The indices N N are unrestricted. Note that the s are disjoint and they partition the set M P. e show that for every Z, and for ; 8< XX< & (13) S S + Equation 12 will follow by summing over all Z. Hence consider a fixed Z. Note that if then! 8< Z. Accordingly we partition into Ä! 5 Z as follows: for!8 ;Z+<, M M Z M + N + + N ;N 8< X ;N 'P N- P " 11

12 K K P For Z we have G< XF Z since for all, N define the sets Ä! ; Z as follows. For! ;Z<, O M M ZN M + N + + N Unlike for ZN, for N Z it could be that N N N are unrestricted in X< & S XX< & S ] ] Hence the claim follows. N X8< N P, so we have so N N = P XJ<. Now given X] let us Z. Since the indices + + G + + G , we have +O + for!8;z. e now prove Equation 13. 8< X < & A two-dimensional greedy array is a matrix of integers such that elements in the same row are congruent mod, while elements in distinct rows are not congruent mod. Lemma 7 says that ordering the elements of column-wise gives a greedy sequence. This concludes the algebraic step of the proof. Let us sketch the rest of the proof when <, which corresponds to the Frankl-ilson construction (see Figure 1). Define the sets M!-P M M A <'P + QÄ! P Q FPP M!-P M M A <'P + QÄ! P ]; FPP Note that!qä! P!Q FP for every! in. Further and partition the set M ;A < 'P and they intersect only at!. e will show that and contain large greedy arrays. 1. Arrange M!N <'P in grid, each row corresponding to a congruence class mod. 2. ithin each row, place elements lying in before those in. Since! lies in, place all other elements in which are!-8b&3 before!. 3. Sort the rows according to how many elements from they contain. This reordering is illustrated in Figure 1, the dark line separates and. t is clear that and contain greedy arrays of size and of size respectively (indicated by shaded regions) so that. From this it follows that ++&++ +"]. Also, we can ensure that! is the last element of these arrays in the column-wise ordering. So Q P is minimized at the last element in, hence by Lemma 7 5"DF ] ++&+ <. Similarly 5"D 7!] + + <, which proves the desired bound. Also, ++/

13 P P Figure 1: Lower bound for < and the Frankl-ilson construction is minimized when M +!- B? P, and! M +!- B3 P (or vice versa), the corresponding polynomials give exactly the Frankl-ilson construction. The proof for general is a high dimensional extension of this argument. The next lemma (Lemma 8) says that any disjoint partition of M!N < 'P into will result in one of the partitions having a greedy array of size. n fact we prove something stronger, we can choose the dimensions of the array to be any solution to Equation 1. e also assume that is of the form which makes it easier to use induction. Lemma 8 Let any positive integers either contains a greedy array of size. Let be disjoint sets of integers such that M!N < 'P. Given satisfying For N. < (1) PROOF: The proof is by induction on. (the dimension of the greedy arrays). hen.-, we have disjoint sets so that M!N or contains a greedy array of size < 'P hence Since < any ordering of and gives greedy arrays of size + 7+ and + + respectively. Given such that, if <, then + +&] <. Assume that the claim is true up to. <. For!8 N <, we define the following sets NF M 7+ N B&3;P N MN < N + N P e define sets N and N similarly. Note that for each N, N and N are disjoint, further N N M!N < 'P. So the induction hypothesis applied to N and N with ; ; implies that either contains a greedy array of size or contains a greedy array of size. e define the following sets ) M N + N has a greedy array 1 M N + N has a greedy array 13 of size of size.

14 P P P Since 8 are disjoint and we have either +1 + ] or +7+8]. Assume +15+8]. e define a greedy array of size as follows. Choose of size. For each NF, the N row of contains the pre-image of in N of dimension. e need to verify that satisfies < >@ &%' MUZ+ N P. Given and, if N, then X B& so the condition holds. Now assume that N, so that N N. Since and are in the same row, Note that &%' MUZ+ N 1N X< P &%')MUZ+ N 8< %8B&3 for!8 % <. So N < - %O < < > X? /%' MUZ + N P. Hence 8< & - %O is a greedy array of the right dimension. The next Lemma is the key step in the combinatorial argument. Now we consider sets and which intersect only at!, and we want to produce greedy arrays that end at! by our ordering. e show that such arrays exist whose dimensions satisfy Equation 1. Lemma Partition Lemma: Let Then there exist positive integers ; of size array contain! as the last element. PROOF: The proof is by induction on.. hen., we have sets so that M!N. Let be sets of integers such that M!N < 'P M!-P satisfying Equation 1, so that contains a greedy and contains a greedy array of size, and both and < 'P and M!-P so +7+" + + e take Define to be an ordering of where! comes last, similarly for.! < Assume that the claim holds up to.f<. For N, we define the sets N N N before. Note that Ä! Ä! M!N Ä! Ä! M!-P <'P. N as By induction, there exist and as above so that Ä! contains a greedy array of size and Ä! contains a greedy array of size. For N < we have N N M!N < 'P N N Hence applying Lemma 8, either N contains an array of size of size. Again we define the sets ) M N + N has a greedy array 1 M N + N has a greedy array of size of size or Ä! contains a greedy array 1

15 / / Let Since / M!-P and M!N A< 'P we have. Order and so that! is the last element. e define a greedy array of size as follows. For each N2, row of the N contains the pre-image of in N of dimension. Similarly we define where the N row contains the pre-image of in N. The proof that these are greedy arrays follows Lemma 8. They both contain! as the last element by induction. e now complete the proof of Theorem. PROOF OF THEOREM : Assume that. e can choose so that and <. Define the sets 1 M!-P M + < QÄ! P ; FPP M!-P M + < QÄ! P ]; FPP Applying Lemma implies that and contain greedy arrays and of size and respectively where and satisfy Equation 1. Applying Lemma 7, by ordering we get a greedy sequence M ; :!-P in of length (. By the definition of set, QÄ! P Q P for N ;. So by Lemma 5"D) "] <. Similarly we get a greedy sequence of length in ending in!. Note that by Equation, and! implies!qä! P!Q FP. So by Lemma, 5"D) 7! ] <. By Equation 1, ] for. < and ]. Hence For the Frankl-ilson construction where 5"D) 5"D 7! ] ] > <, we get 5"DF 5"DF 7! ] which is tight. Lower Bounds for Prime Representations n this section we prove a lower bound for prime representations using symmetric polynomials. /5O / Theorem 1 Let Q " and!q " be symmetric polynomials that represent the OR function on variables. Then 5"D) /5"D 7!"] -"!. Note that this requires 5"D) /5"D 7!] but if 5"D F!, then it is easy to show that 5"D 7! F, so this case is not interesting. The hard case of this theorem is when and ' are fast-growing functions of, as in Alon s construction. To handle this case, we prove a partition lemma (Lemma 11) which says that taking and ' large does not help. / / Definition 1 Let ' be distinct primes, let 8'. Let and. e say that is covered by if B?. e say and cover if every M P is covered by or. f 8', we can cover by taking and. Given / and /, the number of elements in M A8' P that are covered by or is ++ ' + +< which can be much larger than The partition lemma states that to cover the first integers however, needs to be = >. Lemma 11 Partition Lemma: f / and / cover, then,+ 7+O,+ +O >. 15

16 / / / + -,!"/.1"*23!"%)"%("*!'!"$$!"%!"&(' Figure 2: Proof of Prop 12 Using + 7+ rather than + + in the product lets us ignore the case when + 7+K!. Let us sketch the idea behind the proof of the Partition Lemma. Let '. Assume that to begin with, we have and MU'- '- ' P. t is clear that and cover, however,+ 7+,+ + >. One could try and reduce + + by removing elements from it. e want to show that this results in an increase in Removing N- from results in the numbers M N N ' N > < ' P being uncovered. Call this set < N. The various elements of < N are less than 8' and they are congruent mod ', hence the CRT implies they cannot also be congruent mod. But the problem is for N,there could be considerable overlap between the residues of < N and < - mod. Hence it is not clear that removing many elements from does actually cause + + to increase. However, by suitably reordering the elements of /1, we show that every element removed from causes the size of to increase by at least. n fact Figure 2 shows that + + could increase by just. This is sufficient to prove the Partition Lemma. Set. Given set of integers, define / 8B&3 to be the set M 8B&? + P and 5 )7 Proposition 12 Let )] '. f and cover and + + 1'< then + 7+]5 7 <. PROOF: Note that since )];', ]. Let denote the complement of in /, so!(. For each N, take < N to be the first numbers in M P congruent to N B3. n other words, <Ä! MU' '" ' P and for N!N N F M N N '- N) > < ' P. Let );: < < N f, then is not covered by so it must be covered by. e want to lower bound the size of ) B3. Let us reorder the set / as M!N/'- '- (< ' P (this is a reordering since ' A ). t sends = B3 to % - ' such that % - ' = B3. This map sends <Ä! B? to MU'- ; ' P and the set < N B? to the interval M% N '-Ä% N 5 '- ;Ä% N < ' P of length for N+!. None of these intervals contain!, since that would give M ; P such that N B.' and!-8b&3. Such an is not covered by or. Each interval < N 8B&3 begins at a distinct point % N. Sorting the intervals by their starting points, it follows that the union of such intervals of length contains at least "< elements of /. Figure 2 illustrates this argument = /'=. Here * M!NO'P. PROOF OF LEMMA 11: e consider the cases, ;' and ' separately. The non-trivial case is when '. 1

17 1. Let ;'. Numbers M ; P lie in distinct congruence classes mod and '. Hence + 7+O+ +&],++O,+ +" > ++O+ +&] and,+7+o,+ +O >. This proves the claim if so let >. 2. Let '. The numbers M A;P lie in distinct congruence classes modulo and '. Hence Let + + H< for 8. There are numbers in each congruence class mod. Thus numbers are not covered by and have to be covered by. Since ', they lie in distinct congruence classes mod '. Hence + +=]. Using the fact that )] hence ] we get,+ +O,+ +" > < 8-3. Let >;'. By Prop. 12, if + + 1' <, then + 7+] Hence for? < we have,+ 7+O,+ + ] '< FÄ ] > <. Since + 7+ =<, we get =< e will show that this is lower bounded by. By differentiating, this bound is minimized at one of the extreme values of, so it suffices to check the bound is at least for those values. hen $,. hen < 5 7 ',< ' ' ' > '< ' '< ] '< ',< F ' ' N ' One of '< and >NF' is at least : f ', >NF'7>. f '7>, then '< F >. / e now proceed to the algebraic step of the proof. Every symmetric polynomial Q V computes a symmetric function /. Let. Every polynomial R also computes a function /5. The following equivalence between the two kinds of polynomials is given by Theorem 2. of [1]. Proposition 13 The functions that can be computed by symmetric polynomials Q than are the functions which can be computed by polynomials Q ;. = /O of degree less For each variable, let 5"D denote the degree of in. f is the largest index so that 5"D) >! then 5"D). This gives a bound with an error factor of. By defining an appropriate weighted degree of, we can make the correspondence exact. 17

18 Y Y Y Definition 11 Given defined as 5"D ; = /, the degree of a monomial. The degree of with H< is denoted 5"D is the maximum degree over all monomials. Note that if is the largest index such that 5"D >! then 5"D) 8;5"D) ;5"D). / Lemma 1 Given a symmetric polynomial Q there is a unique polynomial / computes the same function and vice versa. This correspondence preserves the degree. /1 PROOF: Given a symmetric multilinear polynomial Q " of degree, write it as Q On at -1 input N GK. By Lucas Theorem B? Further the polynomial has degree. Thus computes the same function as ; R % 8B&? Q R that %. and they have the same degree. To prove the other direction, observe that the monomials with < form a basis for polynomials in /5O with degree at most < in each. Further writing a polynomial in this basis does not change the degree as defined above. Let be the degree of the monomial. Hence given Q with degree, one can write Q ; % By Lucas theorem, this computes the same function as the polynomial Q For M!N ; P, let L ' denote the base and base ' expansions of. For / Q L ;L = L ;L let Q denote the polynomial applied to the base expansion of. As consequence of Lemma 1, to prove Theorem 1 it suffices to prove the following Proposition. Proposition 15 Let Q / " L ;L and! /% $ <U be polynomials such that QÄ! $8B&3 and!8ä! For, Q!, B? or!q Further 5"D JU5"D)!"]. PROOF: Let Z denote the largest index such that 5"DF L =] in Q L ;L. Similarly let [ be the largest index so that 5"D $ B.'!, B&V' % -.. This implies 5"D) +] and Q, ]. Then 5"DF! ] ' Y and!q!8. Hence 5"D 5"D! ] ' Y. This proves the desired bound for "!R ' Y. So we may assume that )] "!R ' Y. Also ' Y, since if ' Y, then L L! and ;! so 8 ' Y QÄ!N :! 18 B&3

19 Y Y Y Y Y Y Y!Q ' Y which contradicts the hypothesis. Let 5 S% T 7 and L as 5"DF L!QÄ!N ;:! 8'. Let us consider inputs of the form ' Y where!8 ' Y B?. Similarly ;! and / H Ä!N ;:!N ' Y, and < as < < and 5"DF 5 15"D) 1;'<. Note that / / e define and. So by Lemma 11, Since 5"D) L /5"D) ] 5"DF! ];' Y 5"D, Ä! H $ B3 and <Ä!!- B3 or to be the! sets of 5"D) L, this implies that 5"DF L 5"D J 5"D! ] 5"D and < 5"D $ B3. Observe that this implies L L! / B.'. Define polynomials + H!8Ä!N ;:!N. This implies 5"D H B.' (15)!- B(' ] 8 ' Y$> The second inequality uses the fact that )] "!R ' Y hence ' Y 5 5 Ramsey Graphs based on Set ntersections respectively. By equation 15 and cover ]. Since 5"D) ] "! > "! S T 7 >. 5"D L The constructions of Frankl-ilson, Alon and Grolmusz use a coloring scheme based on the size of set intersections. n this section we show that these can be constructed from certain polynomials that are closely related to OR polynomials. These polynomials are also used by Grolmusz [18] and Kutin [1] to give simple constructions of set systems with restricted intersections mod. Definition 12 The weight- function is a partial function defined on M!NO'P for ] as follows and The function is undefined for."! if." Note that on variables is simply the NAND function. e now define polynomial representations of. e give an extension of Definition 2, a similar extension holds for Definition / O /$ Definition 13 Polynomials Q " and!8 " V represent the function on variables if Q Q!,8B&? and!8!,8b&? or!8 The degree of the representation is 1V23 5"DF /5"D 7!.!, B&V' if.!, B.' if. Assume that Q and!8 represent with degree for some ]. Define Q and!8 to be polynomial by substituting <? for when N? and setting! for N]. t is easy to verify that and! represent OR on variables with degree at most. Further if and! were symmetric polynomials, then so are and!. Thus lower bounds for OR representations 1

20 imply lower bounds for representations. n particular our lower bounds for OR representations rule out representations of with symmetric polynomials of degree. Conversely one can construct degree representations of from degree symmetric polynomials representing OR on variables. e do not know if a similar statement is true for asymmetric polynomials. Lemma 1 A degree representation of OR on variables using symmetric polynomials gives a degree representation of on variables for all ], PROOF: Let!Q be symmetric polynomials of degree at most on variables that represent OR. Replace each by < and multi-linearize. t is easy to show that the resulting polynomials! represent on variables. Further, they are symmetric multilinear polynomials of degree, hence we can write them as ; Z! [ ; e obtain new polynomials ; Z and! by replacing ;! ; ; by [. ; Since the value of a symmetric function on a!no -input depends only on the weight of the input, one can show that and! represent on variables. e can use the 78 OR representations to construct representations of. e give a construction of explicit Ramsey graphs from representations of. Construction 2 Ramsey Graph 7A5/B from representations of < A consists of vectors M!NO'P of weight.!, add to B. < f Q.. Theorem 17 Given a degree representation of the weight-n function on M!NO'P and, the graph has vertices PROOF: Assume we have prime representation of. e associate a polynomial with each vertex so Q. Given, let! if! and if. Set ( Q ; and multi-linearize. Using an argument like in Theorem 3, we can show that this gives a polynomial representation of over /. Since the s are multilinear polynomials of degree, we get 1. Similarly we bound by representing over /. For prime-power representations of OR, we can represent and over /< S and get a similar bound. The OR representation of Equation 1 gives the following construction due to Alon [2]. Let 8' <. The vertices are all subsets of size. Add to B if < B&. The OR representation of Equation 3 gives the Frankl-ilson construction: Let <. The vertices are all subsets of size. Add edge to B if + :+ +K < B&3. This construction can be extended to."] colors using the polynomials constructed in Equation 5. 2

21 Construction 3 Extending the Frankl-ilson construction to. colors. < Take <. Vertices are all subsets of. < Edges are colored M!N.G< 'P. Edge is given color The OR representation of Equation 2 gives the following graph 8 7A /B. Let <. The vertices are all subsets of size. Add to B if + +X <$ B8. n fact the graph obtained is the same as Grolmusz. To show this, we first present his construction, following the simplified exposition of Grolmusz himself [18] and Kutin [1]. Let <. The BBR polynomials give the following representation of. Q!8 < < /2 Define " to be the polynomial obtained by combining these polynomials using the CRT. t follows that $ B8L when." and is divisible by 2 or 3 when.". e can view as an integer polynomial with coefficients in M!N P. By repeating each monomial sufficiently many times, we can write The elements of the universe are the monomials. f is repeated % times in, then there are % elements in the universe, one for each occurance of. For each M!NO'P of weight, the set < consists of monomials that evaluate to on. One can verify that this system has restricted intersections mod since +1< < +. The vertices of the graph!- B8. e wish to show that this graph is the same as the graph constructed above. e identify ;A with < )AQ. n, we add an edge between < and < if 3@!,8B&Q. By the CRT, this implies that!, BQ. By Lucas theorem, this happens if. < BQ. But this is precisely when is an edge of. This equivalence can also be seen from Kutin s construction [1]. hile Grolmusz s set system construction is an important result, our approach seems to be simpler for the purpose of Ramsey graph construction. One can interpret the bounds on clique and independent set size as coming from an extension of the modular Ray- Chaudhuri-ilson theorem to prime powers, which we prove below (Theorem 18). are the sets <. e add edge < if + < Set Systems with restricted ntersections modulo Prime Powers Definition 1 A set system C M P on is said to have restricted intersections mod ' if there exists so that +1 +8B&.' but +1 + B.'. For a fixed modulus ', we study the problem of how large + C+ can be as a function of. hen 'V is a prime, the non-uniform modular Ray-Chaudhuri ilson theorem proved by Deza, Frankl and Singhi [5] gives a bound of + C+ +.+ < hen ' is not a prime power, Grolmusz shows a lower bound of EHG K [17]. e give a near-tight bound of S due to Babai et al. []. Previously for the prime power case. This improves the bound of 21 /

22 stronger bounds than ours were known for the special case when + + < i.e. when all set sizes are congruent to 8B& for some (see theorems 5.3 and 7.18 in the book by Babai and Frankl [5]). To prove our result, we use the fact that every function from / S to / can be written as a polynomial. Theorem 18 Let C be a set system with restricted intersections modulo. Then + C+N PROOF: e construct a univariate integer-valued polynomial " R By Lucas theorem, Q F < < 5 < < Set Q F B&3 B?!= B&3 B? B&!= B& B&!= B& < < < S. of degree < such that <$8B&3 <$8B&3 $ B3 $ B3 and multi- By Lemma 3.1, of [] this implies the desired bound. e sketch the argument below. e will use to denote the incidence vector of set. Let ; Q linearize. t is easy to show that 1,+1 ++ B? N!, B? N Using this one can show that the polynomials are linearly independent over R. Since they are multilinear polynomials in variables of degree <, the bound follows. Discussion and Open Problems Following the breakthrough of Barak et al. [8], the algebraic construction described here are no longer the best constructions known. However, the appeal of these constructions is their simplicity and elegance, together with the fact that they are very explicit. So we believe that it is important to resolve the question of whether this approach can beat the Frankl-ilson bound. As we have seen, this problem is intimately linked to well-studied questions in complexity theory. Lower Bounds for Asymmetric Polynomials: The question of whether there are lower degree weak representations of the OR function mod has been open for a while. This work raises the question of whether low degree OR representations exist for our definition. Better upper bounds would give better Ramsey graphs. Lower bounds for prime representations will imply lower bounds for weak representations mod. Prime-power representations are exciting from the lower bound viewpoint since they have not been studied previously and might turn out to be easier to work with. The =Ä@CBD lower bound of Barrington and Tardos citebar-tar applies to both kinds of representations. 22