P. Frankl, and R. M. Wilson, "Intersection Theorems with Geometric Consequences", Combinatorica, Vol. 1, pp. 357-368, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....large graphs with small homogeneous vertex sets is a longstanding challenge for combinatorists. The seminal paper of Erdos [2] proved the existence of an O(2 ) vertex graph without a t vertex clique or a t vertex independent set, but the best construction to date due to Frankl and Wilson [3] gives a graph with exp ( 1 Gamma ) log t) log log t vertices. We proved matching bounds in [5] with a method generalizable to explicit Ramsey colorings with more than two colors. In the paper [4] we have found a relation between low rank co diagonal matrices and ....

.... of size n Theta n, using the BBR polynomial of Barrington, Beigel and Rudich [1] An easy computation shows that this matrixconstruction together with Theorem 3 imply an explicit Ramsey graph with homogeneous sets of the same logarithmic order of magnitude as the result of Frankl and Wilson [3]. Now we give another construction for low mod 6 rank co diagonal matrices using modular sieves. This construction is our main result here. 2 Our Construction 2.1 A logarithmic rank co diagonal matrix The first step in the construction is a co diagonal matrix suitable for large moduli. The next ....

P. Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357--368, 1981.

....on the probabilistic method [19, 95, 10] Although it is usually possible to get better bounds with probabilistic method than with constructions, in applications for example, a construction is often preferable. The best constructive lower bound for r(K k , K k ) given by Frankl and Wilson [115] has also important applications, see the section on information theory) The probabilistic method has been used extensively in computer science, see for example the book of Motwani and Raghavan [187] on randomized algorithms) Also Milman s [182] probabilistic proof to Dvoretzky s Ramsey type ....

....Alon [8] disproved this conjecture in a strong sence. He proved that for every k there is a graph G so that c(G) k, c(G) k, while c(G G) 1 o(1) log k) 8 log log k and the o(1) term tends to zero as k tends to infinity. For his proof he used a modified version of Frankl and Wilson s [115] well known ex12 plicit coloring that gives the constructive lower bound k (1 o(1) logk 4loglogk r(K k , K k ) Alon extended to g 2 colors the modification of the Frankl Wilson construction to obtain: 1 o(1) logk) g 1 g g (loglogk) g 1 r(K k , K k , K k ) This ....

P. Frankl and R.M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357--368.

....constructed which does not contain a clique or independent set of size t. Erdos posed as a challenge the problem of constructing a graph whose size is superpolynomial in t and does not contain a clique or independent set of size t. The challenge was answered by Frankl [6] and Frankl and Wilson [7] who showed an explicit way of constructing graphs that are of size t log t . The goal of this paper is to present an elementary construction of a graph whose size is superpolynomial in t (the size we know that there is no subgraph which is a clique or independent set) Our construction is not ....

....an explicit way of constructing graphs that are of size t log t . The goal of this paper is to present an elementary construction of a graph whose size is superpolynomial in t (the size we know that there is no subgraph which is a clique or independent set) Our construction is not as good as [7], the graph size is t for some constant c. A well known conjecture in this area is that the Paley graph is a Ramsey graph. The Paley graph is defined for every prime p which is congruent to 1 mod 4. There is an edge between nodes i and j iff i Gamma j is a quadratic residue mod p. An ....

P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1, 357-368, 1981.

....hypergraph on exp (c(log m) vertices exists. Keywords: set systems, algorithmic constructions, explicit Ramsey graphs, explicit Ramsey hypergraphs 1 Introduction We are interested in set systems with restricted intersection sizes. The famous RayChaudhuri Wilson [RCW75] and Frankl Wilson [FW81] theorems give strong upper bounds for the size of set systems with restricted pairwise intersection sizes. T. Sos asked in 1976 [Sos76] what happens if not the pairwise intersections, but the k wise intersection sizes are restricted. Furedi [Fur83] Fur91] showed (actually proving a much more ....

....we can choose a large enough p for proving the non modular version, p ncertainly su#ces. Our main tool is substituting set systems into multi variate polynomials [Gro01] This tool, together with the linear algebraic proof of Theorem 9 implies our result. In the seminal paper of Frankl and Wilson [FW81], the Frankl Wilson upper bound to the size of a set system was used for an explicit Ramsey graph construction. Similarly, we can also use our Theorem 1 to an explicit construction of a t coloring of the edges of the k uniform complete hypergraph, such that no color class will contain a complete, ....

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Peter Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357--368, 1981.

....on exp (c(log m) vertices exists. Keywords: set systems, algorithmic constructions, explicit Ramsey graphs, explicit Ramsey hypergraphs 1 Introduction We are interested in set systems with restricted intersection sizes. The famous RayChaudhuri Wilson [RCW75] and Frankl Wilson [FW81] theorems give strong upper bounds for the size of set systems with restricted pairwise intersection sizes. T. S os asked in 1976 [S os76] what happens if not the pairwise intersections, but the k wise intersection sizes are restricted. F uredi [F ur83] F ur91] showed (actually proving a much ....

....can choose a large enough p for proving the non modular version, p n certainly suces. Our main tool is substituting set systems into multi variate polynomials [Gro01] This tool, together with the linear algebraic proof of Theorem 9 implies our result. In the seminal paper of Frankl and Wilson [FW81], the Frankl Wilson upper bound to the size of a set system was used for an explicit Ramsey graph construction. Similarly, we can also use our Theorem 1 to an explicit construction of a t coloring of the edges of the k uniform complete hypergraph, such that no color class will contain a complete, ....

[Article contains additional citation context not shown here]

Peter Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981.

....0. Substitute u j in the relation. While j g j (u j ) 6= 0, we have i g i (u j ) 0 for all i 6= j, a contradiction. Remark 3.1. The history of combinatorial applications of the linear algebra bound goes back to Bose [12] 1949) Some of the highlights of this history are [34] 24] 27] [17]. We refer to [5] for a wealth of results obtained through this method. Above we inferred linear independence via the triangular criterion [5, Sec. 2.1.4] 6 LAJOS R ONYAI, L ASZL O BABAI, AND MURALI K. GANAPATHY 4. An improvement Let d = max i d i . In terms of m, n, and d, Theorem 1.1 ....

P. Frankl, R. M. Wilson, Intersection theorems with geometric consequences. Combinatorica 1 (1981) 357-368. 22 LAJOS R  ONYAI, L  ASZL  O BABAI, AND MURALI K. GANAPATHY

....here a construction that supplies a deterministic algorithm to construct a graph with the desired properties in time polynomial in the size of the graph. It is worth noting that for the case r = 2, corresponding to the usual Ramsey numbers, there are several known explicit constructions; see [10] [13], 9] 1] 2] 3] Despite a considerable amount of e ort, all these constructions supply bounds that are inferior to those proved by applying probabilistic arguments. The problem of nding explicit constructions matching the best known bounds is of great interest, and may have algorithmic ....

P. Frankl and R.M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357-368.

....of Babai and Frankl [BF92] covers plenty results related to this topic. Just to mention a few, bounds to the size of set systems with restricted intersections play a main role in the refutation of Borsuk s conjecture [KK93] in results in combinatorial geometry, related to the Hadwiger problem [FW81], and yields the best known explicit Ramseygraphs [FW81] Gro00b] Here we present a method for constructing set systems with prescribed intersections. Most of our results are for constructing set systems with restricted intersections modulo an integer (mostly primes) In Section 4 a by product ....

....to this topic. Just to mention a few, bounds to the size of set systems with restricted intersections play a main role in the refutation of Borsuk s conjecture [KK93] in results in combinatorial geometry, related to the Hadwiger problem [FW81] and yields the best known explicit Ramseygraphs [FW81], Gro00b] Here we present a method for constructing set systems with prescribed intersections. Most of our results are for constructing set systems with restricted intersections modulo an integer (mostly primes) In Section 4 a by product of this method gives new upper bounds for the size of ....

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P'eter Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357--368, 1981.

....Hamming distances must contain few code words. 1 Introduction In this paper we give bounds on the size of set systems and codes, satisfying some k wise intersection size or Hamming distance properties. For k = 2, these theorems were proven by Ray Chaudhuri and Wilson [12] Frankl and Wilson [9], and Delsarte [6] 5] The k 2 case was asked (partially) by T. S os [13] and F uredi [10] proved, that for uniform set systems with small sets, the order of magnitude of the largest set system satisfying k wise or just pair wise intersection constraints are the same (his constant was huge) ....

.... Then jHj (k 1) s X i=0 n i : If in addition the size of every member of H belongs to the set fk 1 ; k t g and k i s t for every i, then jHj (k 1) s X i=s t 1 n i : This theorem has the following modular version, which generalize the theorem of Frankl and Wilson [9] and strengthen the result from [11] Theorem 2 Let p be a prime and L be a subset of f0; 1; p 1g of size s. Let k 2 be an integer and let H be a family of subsets of n element set such that jHj (mod p) 62 L for every H 2 H but jH 1 : H k j (mod p) 2 L for any collection of k ....

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P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica, 1(4):357-368, 1981.

....Hamming distances must contain few code words. 1 Introduction In this paper we give bounds on the size of set systems and codes, satisfying some k wise intersection size or Hamming distance properties. For k = 2, these theorems were proven by Ray Chaudhuri and Wilson [12] Frankl and Wilson [9], and Delsarte [6] 5] The k 2 case was asked (partially) by T. S os [13] and Furedi [10] proved, that for uniform set systems with small sets, the order of magnitude of the largest set system satisfying k wise or just pair wise intersection constraints are the same (his constant was huge) ....

.... i=0 n i : If in addition the size of every member of H belongs to the set fk 1 ; k t g and k i s Gamma t for every i, then jHj (k Gamma 1) s X i=s Gammat 1 n i : This theorem has the following modular version, which generalize the theorem of Frankl and Wilson [9] and strengthen the result from [11] Theorem 2 Let p be a prime and L be a subset of f0; 1; p Gamma 1g of size s. Let k 2 be an integer and let H be a family of subsets of n element set such that jHj (mod p) 62 L for every H 2 H but jH 1 : H k j (mod p) 2 L for any collection ....

[Article contains additional citation context not shown here]

P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica, 1(4):357--368, 1981.

....explicit construction of a Ramsey hypergraph. Keywords: set systems, algorithmic constructions, explicit Ramsey graphs, explicit Ramsey hypergraphs 1 Introduction We are interested in set systems with restricted intersection sizes. The famous RayChaudhuri Wilson [RCW75] and Frankl Wilson [FW81] theorems give strong upper bounds for the size of set systems with restricted pairwise intersection sizes. T. S os asked in 1976 [S os76] what happens if not the pairwise intersections, but the k wise intersection sizes are restricted. Furedi [Fur83] Fur91] showed (actually proving a much ....

....choose a large enough p for proving the non modular version, p n certainly suffices. Our main tool is substituting set systems into multi variate polynomials [Gro01] This tool, together with the linear algebraic proof of Theorem 9 implies our result. In the seminal paper of Frankl and Wilson [FW81], the Frankl Wilson upper bound to the size of a set system was used for an explicit Ramsey graph construction. Similarly, we can also use our Theorem 1 to an explicit construction of a t coloring of the edges of the k uniform complete hypergraph, such that no color class will contain a complete, ....

[Article contains additional citation context not shown here]

P'eter Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357--368, 1981.

....for distance 2, they showed D = n 1 if n 1 0 mod 4, n if n 1 1 mod 4, n 1 if n 1 2 or 3 mod 4. This con guration gave a quadratic lower bound for (IR n ) Erd os conjectured that (IR n ) grows exponentially in n. This conjecture was veri ed by P. Frankl and R. M. Wilson [31], who proved a strong intersection theorem for set systems, which actually generalizes the lemma of Erd os and S os. Using this theorem, P. Frankl and R. M. Wilson exhibited in IR n , in particular among its 0 1 vectors, an (M;D) critical con guration with M=D (1:2 o(1) n . The ....

....for the chromatic numbers of graphs. This framework was actually developed in order to set a tight lower bound for the chromatic number of the Kneser graph, which is the discrete analogue of the Borsuk graph [41] For a smaller , Frankl and Wilson gave exponential lower bound for (S n 1 ) [31]. The measurable analogue of the independence number generalizes for spheres as well. Let m (n 1) H (r) denote the supremum of the independence ratios m (n 1) H (r) sup X S n 1 (r) X) S n 1 (r) 3) where is the (n 1) dimensional Lebesgue measure, S n 1 (r) is the sphere ....

P. Frankl, R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1(4) 1981, 357-368.

....diameter needed to cover X . Consider P d Gamma1 the space of lines through the origin in R d where the metric is given by the angle between two lines. The diameter of P d Gamma1 is =2 and the distance between two lines is =2 iff they are orthogonal. Let d = 4p, p a prime. Frankl and Wilson [17] , see also [39, 16] proved that there are at most 1:8 d vectors in f Gamma1; 1g d such that no two are orthogonal. This yields b(P d Gamma1 ) 1:1 d , since if P d Gamma1 is covered by t sets of smaller diameter, each such set contains at most 1:8 d of the lines spanned by the ....

P. Frankl and R. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 259-286.

....3 and (G) n ffi , for some ffi 1. By selecting k log n in Corollary 2.15, we obtain a Ramsey graph with n O(logn) vertices and no clique or independent set of n vertices. The best explicit construction known for such graphs has size only n O(logn= log log n) See Frankl and Wilson [11]. In fact, the constructions in [11] give the proof to Lemma 3.6. Open question: Is there an efficient deterministic construction of the graphs G k of Problem 2.14 More specifically, for 0 ffi 1, we need to construct a set S ae f1; ng log n such that for some 0 ffl 1 and c ....

.... 1. By selecting k log n in Corollary 2.15, we obtain a Ramsey graph with n O(logn) vertices and no clique or independent set of n vertices. The best explicit construction known for such graphs has size only n O(logn= log log n) See Frankl and Wilson [11] In fact, the constructions in [11] give the proof to Lemma 3.6. Open question: Is there an efficient deterministic construction of the graphs G k of Problem 2.14 More specifically, for 0 ffi 1, we need to construct a set S ae f1; ng log n such that for some 0 ffl 1 and c 0: 1. jSj n ffl log n . 11 2. ....

P. Frankl, R. Wilson. "Intersection theorems with geometric consequences". Combinatorica 1, 357--368, 1981.

.... the logarithmic order of magnitude of the best previously known construction due to Frankl and Wilson (1981) Our construction uses certain mod m polynomials, discovered by Barrington, Beigel and Rudich (1994) 1 Introduction Generalizing the Ray Chaudhuri Wilson theorem [8] Frankl and Wilson [6] proved the following intersection theorem, one of the most important results in extremal set theory: Department of Computer Science, E otv os University, Budapest, Address: R ak oczi ut 5, H 1088 Budapest, HUNGARY; E mail: grolmusz cs.elte.hu. Part of this research was done while the author ....

....s: Then jF j n s : 1) 2 This theorem has numerous applications in combinatorics and in geometry (e.g. the disproof of Borsuk s conjecture by Kahn and Kalai [7] cf. 1] Sec. 5.6. an explicit construction of Ramsey graphs, and geometric applications related to the Hadwiger problem [6]. Frankl and Wilson [6] asked whether inequality (1) remains true when the modulus p is replaced by a composite number m, or at least in the subcase s = m 1. Frankl [5] answered the rst of these questions (arbitrary s m) in the negative: he constructed faster growing set systems for m = 6, ....

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P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), pp. 357-368.

.... they provide lower bounds for the graph Ramsey function) exist [15] It is still an outstanding open question to explicitly construct such graphs; the current best are the breakthroughs of Frankl and Wilson, who constructed (2 O( p log n log log n) 2 O( p log n log log n) n) graphs [17, 18]. Similarly, while nonconstructive progress has been made on the ( off diagonal or Ramsey type ) case of s 6= t [5] where we assume w.l.o.g. that s t, very few constructive results are known; see, e.g. 2] for a construction of (2; O(n 2=3 ) n) graphs. It is known that (2; Theta( p n ....

P. Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1:357--368, 1981.

....with small homogenous vertex sets (i.e. cliques and anti cliques) Consequently, explicitly constructible low rank co diagonal matrices over Z 6 imply explicit Ramsey graph constructions. Our best construction reproduces the logarithmic order of magnitude of the Ramsey graph of Frankl and Wilson [5], continuing the sequence of results on new explicit Ramsey graph constructions of Alon [1] and Grolmusz [6] Our present result, analogously to the constructions of [6] and [1] can be generalized to more than one color. 1 the electronic journal of combinatorics 7 (2000) #R15 2 Our results ....

....a homogeneous vertex set of size 2 c 0 p log n log log n , for some c 0 0, or in other words, an explicit Ramsey graph construction on 2 c 00 log 2 t log log t vertices, without homogeneous vertex set of size t, for some c 00 0. This bound was first proven by Frankl and Wilson [5] with a larger (better) constant than our c 00 , using the famous Frankl Wilson theorem [5] We also gave a construction, using the BBR polynomial [3] and also the Frankl Wilson theorem in [6] A generalization of our main result for ring Zm , where m has more than two prime divisors: Theorem 9 ....

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P. Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357--368, 1981.

.... in combinatorics, Erdos [7] proved that R(m; m) Omega Gamma m2 m=2 ) The problem of finding explicit edge colorings yielding a similar estimate is still open, despite a considerable amount of efforts by various researchers, and the best known explicit construction is due to Frankl and Wilson [9], who gave an explicit 2 edge coloring of the complete graph on m (1 o(1) log m 4 log log m vertices with no monochromatic clique on m vertices. See also [11] 2] for some multi colored variations. These constructions do not supply any nontrivial explicit lower bounds for R(s; m) where s ....

P. Frankl and R. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 259--286.

....L = f0; 1; k Gamma 1g and F [ i=k X i . Then jF j jX k j. The equality holds if and only if F = X k . Next, we show that some modular versions of Ray Chaudhuri Wilson Theorem [7] also extend to polynomial semi lattices. First the uniform case (Frankl and Wilson s modular version [5]) Theorem 5. Let (X; be a polynomial semi lattice of height n, s, k 2 N with s k, L I n [f0g and F X k an L intersection family. Suppose 0 ; 1 ; Delta Delta Delta ; s are distinct residues modulo a prime p such that m k j 0 (mod p) and 8l 2 L; m l j i (mod p) for some i; 1 i s. ....

P. Frankl and R. M. Wilson, "Intersection Theorem with Geometric Consequences, " Combinatorica 1(4) (1981) 357-368.

....of height n, k N, an L intersection family for L = 1, k 1 and F## i=k X i . Then . X k . Next, we show that some modular versions of Ray Chaudhuri Wilson Theorem [7] also extend to polynomial semi lattices. First the uniform case (Frankl and Wilson s modular version [5] ) Theorem 5. Let (X,#) be a polynomial semi lattice of height n, s, k N with k, L # 0 k an L intersection family. Suppose 0 , 1 , s are distinct residues modulo a prime p such that m k 0 (mod p) and i (mod p) for some i, 1 s. Further suppose that for every i , #l ....

P. Frankl and R. M. Wilson, "Intersection Theorem with Geometric Consequences," Combinatorica 1(4) (1981) 357-368.

....s01 l s01 0 l s02 k 0 l s l s 0 l s01 (12) hold. Reduction 11 ( 4] ff(k; L) maxfff(k; L 0 flg) ff(l; L [0; l 0 1] ff(k 0 l; L 0 l)g for all 0 l 2 L. 13) Reduction 12 ( 5] ff(k; L) maxfff(k; L 0 flg) k 0 l) ff(l; L (L 0 (k 0 l) g for all 0 l 2 L. 14) Reduction 13 ([9]) Suppose q is a power of the prime p. Let 0 ; s be distinct residues modulo q. Suppose F ae 0 [n] k 1 satisfies k j 0 (mod q) and if F; F 0 2 F , F 6= F 0 then jF F 0 j j i (mod q) for some 1 i s. If there exists an integer valued polynomial g(x) of degree d such that ....

P. Frankl, R.M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1:357--368, 1981.

....and Singhi [4] proved that for q = p a prime, m(n; s; p) n s ; 1) under the additional hypothesis that, for all E 2 F , jEj 0 (mod p) for some 0 2 N q n L. This hypothesis was removed by Alon, Babai, and Suzuki in [1] This result followed the paper by Frankl and Wilson [5] which established the bound jF j n s for set systems that are L intersecting mod p and 0 uniform, i.e. jEj = 0 for all E 2 F . Note that the right hand side of inequality (1) is less than n s . Frankl was the rst to prove that inequality (1) does not extend to nonprime moduli. ....

....we then construct in Sections 4 and 5. In Sections 6 and 7 we obtain tight log asymptotic bounds for the degrees of separating polynomials in terms of s and k; we also prove that 2 s 1 is tight if we want a bound independent of k. In Section 8, we use higher incidence matrices, following [9] and [5], to obtain a stronger version of Theorem 1.2. In Section 9, we examine the special case L = f0; 1; s 1g; in this case, we can improve the upper bound to m n 2s . Finally, in Section 10, we list open questions. 2. DEFINITIONS 6 BABAI, FRANKL, KUTIN, STEFANKOVI C Throughout ....

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P. Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1:357-368, 1981.

.... : s g is a collection of non negative integers such that 0 1 2 : s ; then m(n; S) X ijSj n i : Frankl and Wilson, using a rank argument weakened the hypothesis in the uniform Ray Chaudhuri Wilson theorem and proved their celebrated modular Ray ChaudhuriWilson theorem [8]. Later, Alon, Babai and Suzuki using another argument proved the Frankl Ray Chaudhuri Wilson theorems and their generalisations for other lattices as well. Even though these are powerful theorems, the proof techniques used by Frankl Wilson and Alon Babai Suzuki do not always give the best result. ....

P. Frankl and R.M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357--368, 1981.

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P. Frankl, and R. M. Wilson, "Intersection Theorems with Geometric Consequences", Combinatorica, Vol. 1, pp. 357-368, 1981.

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P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences. Combinatorica 1, (1981), 259-286.

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