n 1 conv(ai) 0. ( 8. 1 ) we get u1 = u2 = = ur. Hence the common value of all the Uj Tverberg's Tl1eorem

Transcription

1 8.3 Tverberg's Tl1eorem 203 hence Uj E cone(aj ) Above we have erive L;=l 'Pi (uj ) = 0, an so by ( 8. 1 ) we get u1 = u2 = = ur. Hence the common value of all the Uj belongs to n;=l cone(aj ). It remains to check that Uj =/= 0. Since we assume 0 conv(a), the only nonnegative linear con1bination of points of A equal to 0 is the trivial one, with all coefficients 0. On the other han, since not all the ai are 0, at least one Uj is expresse as a nontrivial linear combination of points of A. This D proves Proposition an Tverberg's theorem as well. The colore Tverberg theorem. If we have 9 points in the plane, 3 of them re, 3 blue, an 3 white, it turns out that we can always partition them into 3 triples in such a way that each triple has one re, one blue, an one white point, an the 3 triangles etermine by the triples have a nonempty intersection. The colore Tverberg theorem is a generalization of this statement for ar bitrary an r. We will nee it in Section 9.2, for a result about many simplices with a common point. In that application, the colore version is essential (an Tverberg's theorem alone is not sufficient) Theorem (Colore Tverberg theorem). For any integers r, > 2 there exists an integer t such that given any t ( +1 ) -point set Y C R par titione into + 1 color classes Y,..., Y+ l with t points each, there exist r pairwise isjoint sets A 1,..., Ar such that each Ai contains exactly one point of each lj, j = 1, 2,..., + 1 (that is, the Ai are rainbow), an n 1 conv(ai) 0. Let Tcol (, r) enote the s1nallest t for which the conclusion of the theorem hols. It is known that Tcoi (2, r) = r for all r. It is possible that Tcol (, r) = r for all an r, but only weaker bouns have been prove. The strongest known result guarantees that Tcol (, r) < 2r-1 whenever r is a prime power. Recall that in Tverberg's theorem, if we nee only the existence of T(, r), rather than the precise value, several simple arguments are available. In con trast, for the colore version, even if we want only the existence of Tcol (, r), there is essentially only one type of proof, which is not easy an which uses topological methos. Since such methos are not consiere in this book, we have to omit a proof of the colore Tverberg theorem.. Bibliography an remarks. Tverberg's theorem was conjecture by Birch an prove by Tverberg (really!) [Tve66]. His original proof is

2 210 Chapter 9: Geon1etric Selection Theorerns a E R is containe in more than n - l hyperplanes of 1i. Consequently, at most O ( n ) X-simplices have a on their bounary. Proof. For each -tuple S whose hyperplane contains a, we choose an inclusion-minimal set K(S) C S whose affine hull contains a. We claim that if IK(SI ) I = I K(S2 ) I = k, then either K(S1 ) = K(S2 ) or K(S1 ) an K(S2) share at most k-2 points. Inee, if K(S1 ) = { xi,..., X k - 1, xk } an K(S2) = {x 1,..., X k -1, Yk }, X k =I= Yk, then the affine hulls of K(S1 ) an K(S2) are istinct, for otherwise, we woul have k+1 points in a common (k-1)-flat, contraicting the general position of X. But then the affine hulls intersect in the (k-2)-flat generate by x 1,..., X k - 1 an containing a, an K(S1 ) an K(S2 ) are not inclusion minimal. Therefore, the first k - 1 points of K ( S) etermine the last one uniquely, an the number of istinct sets of the form K(S) of carinality k is at most n k - l. The number of hyperplanes etermine by X an containing a given k-point set K C X is at most n - k an the leinina follows by suinining o over k., Bibliography an remarks. The planar version of the first selec tion lemma, with the best possible constant, was prove by Boros an Fiirei [BF84). A generalization to an arbitrary imension, with the first of the two proofs given above, was foun by Barany [Bar82]. The iea of the proof of Lemma was communicate to me by Janos Pach. Boros an Fiirei [BF84] actually showe that any centerpoint of X works; that is, it is containe in at least ( ) X-triangles. Wag ner an Welzl (private communication) observe that a centerpoint works in every fixe imension, being common to at least c ( 1 ) X-simplices. This follows from known results on the face numbers of convex polytopes using the Gale transform, an it provies yet another proof of the first selection lemma, yieling a slightly better value of the constant c than that provie by Barany's proof. Moreover, for a centrally symmetric point set X this metho implies that the origin is containe in the largest possible number of X-simplices. As for lower bouns, it is known that no n-point X c R in gen eral position has a point common to more than 21 ( 1 ) X-simplices (Bar82]. It seems that suitable sets might provie stronger lower bouns, but no results in this irection are known The Secon Selection Lemma In this section we continue using the term X-simplex in the sense of Sec tion 9.1; that is, an X-simplex is the convex hull of a (+ 1 )-point subset

3 9.2 The Secon Selection Lemma 21 1 of X. In that section we saw that if X is a set in R an we consier all the X-simplices, then at least a fixe fraction of them have a point in common. What if we o not have all, but many X-simplices, some a-fraction of all? It turns out that still many of them must have a point in common, as state in the secon selection lemma below Theorem {Secon selection lemma). Let X be an n-point set in R an let :F be a famil.r of a ( 1 ) X-simplices, where a E (0, 1] is a parameter. Then there exists a point containe in at least X -simplices of :F, where c = c( ) 0 an s are constants. This result is alreay interesting for a fixe. But for the application that n1otiva.te the iscovery of the secon selection lemma, namely, trying to boun the number of k-sets (see Chapter 11), the epenence of the boun on a is important, an it woul be nice to etermine the best possible values of the exponent s. For = 1 it is not too ifficult to obtain an asymptotically sharp boun (see Exercise 1). For = 2 the best known boun (probably still not sharp) is as follows: If IFI = n3 - v, then there is a point containe in at least 0( n3-3 v / log5 n) X-triangles of :F. In the parameterization as in The orem , this means that s 2 can be taken arbitrarily close to 3, provie that a is sufficiently small, say a < n - 8 for some 6 > 0. For higher imen sions, the best known proof gives s :::::: (4+ 1 ) +l. Hypergraphs. It is convenient to formulate some of the subsequent con sierations in the language of hypergraphs. Hypergraphs are a generalization of graphs where eges can have more than 2 points (from another point of view, a hypergraph is synonymous with a set system). A hypergraph is a pair H = (V, E), where V is the vertex set an E C 2 v is a system of subsets of V, the ege set. A k-uniform hypergraph has all eges of size k (so a graph is a 2-uniform hypergraph). A k-partite hypergraph is one where the vertex set can be partitione into k subsets vl ' v2 '... ' vk ' the classes, so that each ege contains at most one point from each Vi. The notions of subhypergraph an isomorphism are efine analogously to these for graphs. A subhypergraph is obtaine by eleting some vertices an some eges (all eges containing the elete vertices, but possibly more). An isomorphism is a bijection of the vertex sets that maps eges to eges in both irections (a renaming of the vertices). Proof of the secon selection lemma. The proof is somewhat similar to the secon proof of the first selection lemma (Theorem 9.1.1). We again use the fractional Helly theorem. We nee to show that many ( + 1 )-tuples of X-simplices of :F are goo (have nonempty intersections). >

4 212 Chapter 9: Geon1etric Selection Theorerns We can view F as a ( +1 ) -uniform hypergraph. That is, we regar X as the vertex set an each X-simplex correspons to an ege, i.e., a subset of X of size + 1. This hypergraph captures the "combinatorial type" of the family F, an a specific place1nent of the points of X in R then gives a concrete "geometric realization" of F. First, let us concentrate on the simpler task of exhibiting at least one goo ( +1 )-tuple; even this seems quite nontrivial. Why cannot we procee as in the secon proof of the first selection lemma? Let us give a concrete example with 2. Following that proof, we woul consier 9 points in R2, an Tverberg's theorem woul provie a partition into triples with intersecting convex hulls: == But it can easily happen that one of these triples, say {a, b, c}, is not an ege of our hypergraph. Tverberg's theorem gives us no aitional information on which triples appear in the partition, an so this argument woul guarantee a goo triple only if all the triples on the consiere 9 points were containe in F. Unfortunately, a 3-uniform hypergraph on n vertices can contain more than half of all possible ( ) triples without containing all triples on some 9 points (even on 4 points ). This is a "higher-imensional" version of the fact that the complete bipartite graph on + vertices has about! n2 eges without containing a triangle. Hypergraphs with many eges nee not contain complete hypergraphs, but they have to contain complete multipartite hypergraphs. For example, a graph on n vertices with significantly more than n31 2 eges contains K2, 2, the complete bipartite graph on vertices ( see Section 4.5). Concerning hypergraphs, let K+ 1 (t) enote the co1nplete ( +l )-partite ( +l ) -uniform hypergraph with t vertices in each of its + 1 vertex classes. The illustration shows a K3 (4); only three eges are rawn as a sample, although of course, all triples connecting vertices at ifferent levels are present. If t is a constant an we have a ( +l ) -uniform hypergraph on n vertices with sufficiently many eges, then it has to contain a copy of K+ 1 ( t) as a subhypergraph. We o not formulate this result precisely, since we will nee a stronger one later.

5 9.2 The Secon Selection Lemma 2 13 In geometric language, given a family :F of sufficiently many X-simplices, we can color some t points of X re, some other t points blue,..., t points by color ( +l ) in such a way that all the rainbow X-simplices on the (+ 1 ) t colore points are present in :F. An in such a situation, if t is a sufficiently large constant, the colore Tverberg theorem ( Theorem 8.3.3) with r = +l claims that we can fin a ( +1 ) -tuple of vertex-isjoint rainbow X-simplices whose convex hulls intersect, an so there is a goo ( + 1 ) -tuple! In fact, these are the consierations that le to the formulation of the colore Tverberg theorem. For the fractional Helly theorem, we nee not only one but many goo ( + 1 ) -tuples. We use an appropriate stronger hypergraph result, saying that if a hypergraph has enough eges, then it contains many copies of K+I ( t) : Theorem (The Eros-Simonovits theorem). Let an t be pos itive integers. Let 1-l be a ( +l ) -uniform hypergraph on n vertices an with a ( 1 ) eges, where a > Cn - l ft for a certain sufficiently large constant C. Then 1-l contains at least copies of K+I ( t ), where c = c (, t ) > 0 is a constant. For completeness, a proof is given at the en of this section. Note that in particular, the theorem implies that a ( +l ) -uniform hy pergraph having at least a constant fraction of all possible eges contains at least a constant fraction of all possible copies of K+I (t). We can now finish the proof of the secon selection lemma by ouble counting. The given family F, viewe as a ( +1 ) -uniform hypergraph, has t+ I n<+i)t copies of K+ l ( t ) a ( 1 ) eges, an thus it contains at least ca by Theorem As was explaine above, each such copy contributes at least one goo ( +l ) -tuple of vertex-isjoint X-simplices of F. On the other han, + 1 vertex-isjoint X-simplices have together ( + 1 ) 2 vertices, an hence their vertex set can be extene to a vertex set of some K+ t ( t ) ( which has t(+1) vertices ) in at most nt ( + l ) -( + l )2 = n<t - - l ) ( +I ) ways. This is the rnaxirnurn nurnber of copies of K+I ( t ) that can give rise to the same t +l ( +I ) 2 n goo ( + l ) -tuples goo ( +1 ) -tuple. Hence there are at least ca +I of X-simplices of :F. By the fractional Helly theorem, at least c' at n+ l X -simplices of :F share a common point, with c' = c' ( ) > 0. This proves the D secon selection lemma, with the exponent s < ( 4+ l ) + t. Proof of the Eros-Simonovits theorem (Theorem 9.2.2). By inuc tion on k, we are going to show that a k-uniform hypergraph on n vertices an with m eges contains at least fk ( n, m ) copies of K k ( t), where

6 2 14 Chapter 9: Geometric Selection Tl1eoren1s with ck > 0 an Ck suitable constants epening on k an also on t (t is not shown in the notation, since it remains fixe). This claim with k = + 1 implies the Eros-Simonovits theorem. For k = 1, the claim hols. So let k > 1 an let 1i be k-uniform with vertex set V, lvi = n, an ege set E, lei = m. For a vertex v E V, efine a (k- 1 )-uniform hypergraph Hv on V, whose eges are all eges of 1l that contain v, but with v elete; that is, 1-lv = ( V, {e \ {v}: e E E, v E e}). Further, let tl' be the (k-1)-uniform hypergraph who e ege set is the union of the ege sets of all the 1iv. Let /C enote the set of all copies of the complete (k- 1 )-partite hyper graph K k - 1 ( t) in 1i'. The key notion in the proof is that of an extening vertex for a copy K E K: A vertex v E V is extening for a K E K if K is containe in 1-lv, or in other wors, if for each ege e of K, eu{ v} is an ege in 1l. The picture below shows a K2(2) an an extening vertex for it (in a 3- regular hypergraph). The iea is to count the number of all pairs (K, v ), where K E K an v is an extening vertex of K, in two ways. On the one han, if a fixe copy K E K has QK extening vertices, then it contributes (qf ) istinct copies of K k (t) in 11.. We note that one copy of Kk (t) comes from at most 0 ( 1 ) istinct K E K in this way, an therefore it suffices to boun L: KEK (qf) from below. On the other han, for a fixe vertex v, the hypergraph 1-lv contains at least fk - 1 ( n, mv) copies K E K by the inuctive assumption, where mv is the number of eges of Hv. Hence L QK > L fk - 1 (n, mv) KEK v EV Using E v EV mv = km, the convexity of fk - 1 in the secon variable, an Jensen's inequality (see page xvi), we obtain L QK > n fk - l (n, km/n). (9. 1 ) KEK To conclue the proof, we efine a convex function extening the binomial coefficient ( ) to the omain R: g(x } = { (x - 1 ).t!. (x - t+ l ) for x < for x > t - 1, t-1.

7 9.3 Orer Types an the San1e-Type Lemn1a 215 We want to boun K E JC g (QK) from below, an we have the boun (9.1) for K E JC QK Using the boun IKI < nt ( k - 1) (clear, since K k - 1 ( t ) has t(k-1) vertices) an Jensen's inequality, we erive that the number of copies of K k ( t ) in 1-l is at least n fk - 1 (n, kmjn ) ) -1 t(k. en g nt ( k - 1) 0 A calculation finishes the inuction step; we omit the etails. ( ) The secon selection lemma was conjecture, an prove in the planar case, by Baniny, Fiirei, an Lovasz [BFL90]. The missing part for higher imensions was the col ore Tverberg theorern (iscusse in Section 8.3). A proof for the planar case by a ifferent technique, with consierably better quanti tative bouns than can be obtaine by the metho shown above, was given by Aronov, Chazelle, Eelsbrunner, Guibas, Sharir, an Wenger [ACE+ 91] (the bouns were mentione in the text). The full proof of the secon selection lemma for arbitrary i1nension appears in Alon, Barany, Fiirei, an Kleitman [ABFK92]. Several other "selection lemmas," sometimes involving geometric objects other than simplices, were prove by Chazelle, Eelsbrunner, Guibas, Herschberger, Seiel, an Sharir [CEG + 94]. Theorem is from Eros an Sirnonovits [ES83]. Bibliography an remarks. Exercises 1. (a) Prove a one-imensional selection lemma: Given an n-point set X C R an a family F of a ( ) X-intervals, there exists a point common to n( a2 ( ) ) intervals of :F. What is the best value of the constant of proportionality you can get? m (b) Show that this result is sharp (up to the value of the multiplicative constant) in the full range of a. 2. (a) Show that the exponent 8 2 in the secon selection lemma in the plane cannot be smaller than 2. (b) Show that 8 3 > 2. m Can you also show that S > 2? ( c) Show that the proof metho via the fractional Helly theorem cannot give a better value of s2 than 3 in Theorem That is, construct an n-point set an a ( ) triangles on it in such a way that no more than O(a5n9) triples of these triangles have a point in common. 9.3 Orer Types an the Same-Type Lemma There are infinitely many 4-point sets in the plane in general position, but there are only two "combinatorially istinct" types of such sets: The orer type of a set.