Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit ramsey graphs. Combinatorica, 20(1):71-86, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 is fixed and m grows. Such constructions for s = 3 appear in various papers, see [1] 6] where it is shown that R(3; m) Omega Gamma m 3=2 ) via an explicit ....

V. Grolmusz, Superpolynomial size set systems with restricted intersections mod 6 and explicit Ramsey graphs, Combinatorica, to appear. (Preliminary version in: Lecture Notes in Computer Science Vol. 1276, 1997, 82-90.)

No context found.

V. Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73--88, 2000.

No context found.

V. Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73--88, 2000.

....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 Ramsey graphs. Definition 1 ( 4] Let R be a ring and let n be a positive integer. We say, that the n Theta n matrix A = fa ....

....let i and j be chosen so that a ij = s, then there are two cases. 6 Case 1: s 2 . 1: 5) In Case 1 there are no problems, for an arbitrary modulus, c ij is non 0. Case 2: s 2 . Gamma 1 : 6) At this point we need a simple Lemma, its proof can be found e.g. in [5]. Lemma 6 Let p be a prime, k, j, e non negative integers, e 1. For any k p , j p (mod p) Now we deal with Case 2. Let s = s mod 2 , that is, 0 s 2 , s j s (mod 2 ) From (6) c ij j (mod 2) 7) That means that c ij j 0 (mod 2) if s ....

V. Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73--88, 2000.

.... ij j 1 (mod p k ) Clearly, a (bilinear) Sigma Pi Sigma circuit compute an a strong representation of polynomial n (x# y) if and only if the corresponding rectangle cover satisfies Properties (a) and (b) The construction of suchalow cardinality rectangle cover is implicit in papers [9] and [10]. We present here a short direct proof which is easily generalizable for proving the results in the next section for higher dimensional matrices. Rectangles, covering M , will be denoted R(I#J) x i ) j2J y j ) We define now an initial cover of the non diagonal elements of M by ....

.... z 2f0# 1g , which contains at most d 1 s: f d# (z) j 0 (mod m) z =0# (ii) If f d# (z) 6j 0 (mod m) then there exists i 2f1# 2#: #rg: f d# (z) j 1 (mod p i ) and if f d# (z) 6j 1 (mod p j ) thenf d# (z) j 0 (mod p j ) Proof: The proof of part (i) is given in [2] seealso[10]) The proof of part (ii) We consider m = p r to be a constant. Let us define q i = m=p i , and let q i j 1 (mod p i ) for i =1# 2#: #r. Let w denote the polynomial satisfying the requirements of (i) Suppose first that e k =1fork =1# 2#: #r. Then w clearly satisfies ....

V. Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73--88, 2000.

.... explicit construction of a t coloring of the edges of the k uniform complete hypergraph, such that no color class will contain a complete, monochromatic hypergraph on a vertex set of size exp(c(log n log log n) Our explicit construction is similar to the explicit Ramsey graph construction of [Gro00]. We note, that much better explicit Ramsey hypergraphs can be constructed using the Steppingup Lemma of Erdos and Hajnal [GRS80] from an explicit construction of k uniform hypergraphs a (much larger) explicit construction of k 1 uniform hypergraphs follows, where k 3. Another construction ....

....) dimensional vector space V . By Theorem 7, hypergraph f(H) satisfies the assumptions of Theorem 9, so f(H) #(k 3 Set systems with restricted k wise intersections In this section we give an explicit construction for a set system with similar (but stronger) properties described in [Gro00]. It was conjectured (see [BF92] that if is a set system over an n element universe, satisfying that #H #H: H #0 (mod 6) but #G, #H,G#= H : G # H ##0 (mod 6) has size polynomial in n. The conjecture was motivated by theorems of Frankl and Wilson, showing polynomial upper bounds for ....

[Article contains additional citation context not shown here]

Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73--88, 2000.

.... explicit construction of a t coloring of the edges of the k uniform complete hypergraph, such that no color class will contain a complete, monochromatic hypergraph on a vertex set of size exp(c(log n log log n) Our explicit construction is similar to the explicit Ramsey graph construction of [Gro00]. We note, that much better explicit Ramsey hypergraphs can be constructed using the Steppingup Lemma of Erd os and Hajnal [GRS80] from an explicit construction of k uniform hypergraphs a (much larger) explicit construction of k 1 uniform hypergraphs follows, where k 3. Another construction ....

....vector space V . By Theorem 7, hypergraph f(H) satis es the assumptions of Theorem 9, so jHj = jf(H)j (k 1) 1 A : 3 Set systems with restricted k wise intersections In this section we give an explicit construction for a set system with similar (but stronger) properties described in [Gro00]. It was conjectured (see [BF92] that if H is a set system over an n element universe, satisfying that 8H 2 H: jHj 0 (mod 6) but 8G; H 2 H; G 6= H : jG Hj 6 0 (mod 6) has size polynomial in n. The conjecture was motivated by theorems of Frankl and Wilson, showing polynomial upper bounds ....

[Article contains additional citation context not shown here]

Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73-88, 2000.

....s = 1 (q p) 2. While this shows that in these cases, m(n; s; q)n s 1 as n 1 (q; s xed) the rate of growth is still polynomial, i.e. of the form O(n c(s) 2) for some function c(s) The value c(s) does not depend on q or n. In fact, in these examples, c(s) 2s. Recently Grolmusz [7] proved that much larger set systems satisfying the conditions exist if the modulus is an integer which is not a prime power: in this case, 4 BABAI, FRANKL, KUTIN, STEFANKOVI C m(n; q 1; q) exp C(q) log n) r (log log n) r 1 ; 3) where r is the number of distinct prime ....

V. Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20(1):71-86, 2000.

....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 set systems ....

....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 set systems with restricted intersections. Surprisingly, this upper bound together with the construction of [Gro00b] can be used for giving lower bounds for the degree (or weight) of some mod 6 polynomials (see Corollary 29 (cf. BBR94] TB98] Gro95] 1.1 Set systems with prescribed intersections We are interested in the following Problem 1 There are given non negative integers a ij ; 1 i j m. Does ....

[Article contains additional citation context not shown here]

Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73--88, 2000.

.... construction of a t coloring of the edges of the k uniform complete hypergraph, such that no color class will contain a complete, monochromatic hypergraph on a vertex set of size exp(c(log n log log n) 1=t ) Our explicit construction is similar to the explicit Ramsey graph construction of [Gro00]. By our knowledge, this is the first explicit Ramsey hypergraph construction in the literature. Grolmusz: Restricted Multiple Intersections Ramsey Hypergraphs 2 2 Preliminaries We define the dream product of matrices of same dimensions, where the product of two matrices is a matrix with each ....

.... Theorem 7, hypergraph f(H) satisfies the assumptions of Theorem 9, so jHj = jf(H)j (k Gamma 1) 0 jLj X i=0 n i 1 A : 2 3 Set systems with restricted k wise intersections In this section we give an explicit construction for a set system with similar properties described in [Gro00]. It was conjectured (see [BF92] that if H is a set system over an n element universe, satisfying that 8H 2 H: jHj j 0 (mod 6) but 8G;H 2 H; G 6= H : jG Hj 6j 0 (mod 6) has size polynomial in n. The conjecture was motivated by theorems of Frankl and Wilson, showing polynomial upper bounds ....

[Article contains additional citation context not shown here]

Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:73--88, 2000.

....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 give a recipe for constructing explicit Ramsey graphs from explicit low rank co diagonal matrices over Z 6 , ....

....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 give a recipe for constructing explicit Ramsey graphs from explicit low rank co diagonal matrices over Z 6 , analogously to the way that our results gave a method for ....

[Article contains additional citation context not shown here]

V. Grolmusz. Superpolynomial size set systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20:1--14, 2000. Conference version appeared in Proc. COCOON'97, LNCS 1276.

No context found.

Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit ramsey graphs. Combinatorica, 20(1):71-86, 2000.

No context found.

Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit ramsey graphs. Combinatorica, 20(1):71-86, 2000.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC