36 research outputs found

    Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices

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    A set system is L-intersecting if any pairwise intersection size lies in L, where L is some set of s nonnegative integers. The celebrated Frankl-Ray-Chaudhuri-Wilson theorems give tight bounds on the size of an L-intersecting set system on a ground set of size n. Such a system contains at most (ns)\binom{n}{s} sets if it is uniform and at most i=0s(ni)\sum_{i=0}^s \binom{n}{i} sets if it is nonuniform. They also prove modular versions of these results. We consider the following extension of these problems. Call the set systems A1,,Ak\mathcal{A}_1,\ldots,\mathcal{A}_k {\em L-cross-intersecting} if for every pair of distinct sets A,B with AAiA \in \mathcal{A}_i and BAjB \in \mathcal{A}_j for some iji \neq j the intersection size AB|A \cap B| lies in LL. For any k and for n > n 0 (s) we give tight bounds on the maximum of i=1kAi\sum_{i=1}^k |\mathcal{A}_i|. It is at most max{k(ns),(nn/2)}\max\, \{k\binom{n}{s}, \binom{n}{\lfloor n/2 \rfloor}\} if the systems are uniform and at most max{ki=0s(ni),(k1)i=0s1(ni)+2n} \max\, \{k \sum_{i=0}^s \binom{n}{i} , (k-1) \sum_{i=0}^{s-1} \binom{n}{i} + 2^n\} if they are nonuniform. We also obtain modular versions of these results. Our proofs use tools from linear algebra together with some combinatorial ideas. A key ingredient is a tight lower bound for the rank of the inclusion matrix of a set system. The s*-inclusion matrix of a set system A\mathcal{A} on [n] is a matrix M with rows indexed by A\mathcal{A} and columns by the subsets of [n] of size at most s, where if AAA \in \mathcal{A} and B[n]B \subset [n] with Bs|B| \leq s, we define M AB to be 1 if BAB \subset A and 0 otherwise. Our bound generalizes the well-known result that if A<2s+1|\mathcal{A}| < 2^{s+1}, then M has full rank A|\mathcal{A}|. In a combinatorial setting this fact was proved by Frankl and Pach in the study of null t-designs; it can also be viewed as determining the minimum distance of the Reed-Muller codes

    Minimum saturated families of sets

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    We call a family F\mathcal{F} of subsets of [n][n] ss-saturated if it contains no ss pairwise disjoint sets, and moreover no set can be added to F\mathcal{F} while preserving this property (here [n]={1,,n}[n] = \{1,\ldots,n\}). More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an ss-saturated family of subsets of [n][n] has size at least (12(s1))2n(1 - 2^{-(s-1)})2^n. It is easy to show that every ss-saturated family has size at least 122n\frac{1}{2}\cdot 2^n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2+ε)2n(1/2 + \varepsilon)2^n, for some fixed ε>0\varepsilon > 0, seems difficult. In this note, we prove such a result, showing that every ss-saturated family of subsets of [n][n] has size at least (11/s)2n(1 - 1/s)2^n. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on F1++Fs|\mathcal{F}_1| + \ldots + |\mathcal{F}_s| where F1,,Fs\mathcal{F}_1, \ldots, \mathcal{F}_s are families of subsets of [n][n], such that there are no ss pairwise disjoint sets, one from each family Fi\mathcal{F}_i, and furthermore no set can be added to any of the families while preserving this property. We show that F1++Fs(s1)2n|\mathcal{F}_1| + \ldots + |\mathcal{F}_s| \ge (s-1)\cdot 2^n, which is tight e.g.\ by taking F1\mathcal{F}_1 to be empty, and letting the remaining families be the families of all subsets of [n][n].Comment: 8 page

    Exact k -Wise Intersection Theorems

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    A typical problem in extremal combinatorics is the following. Given a large number n and a set L, find the maximum cardinality of a family of subsets of a ground set of n elements such that the intersection of any two subsets has cardinality in L. We investigate the generalization of this problem, where intersections of more than 2 subsets are considered. In particular, we prove that when k−1 is a power of 2, the size of the extremal k-wise oddtown family is (k−1)(n− 2log2(k−1)). Tight bounds are also found in several other basic case

    Sperner systems with restricted differences

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    Let F\mathcal{F} be a family of subsets of [n][n] and LL be a subset of [n][n]. We say F\mathcal{F} is an LL-differencing Sperner system if ABL|A\setminus B|\in L for any distinct A,BFA,B\in\mathcal{F}. Let pp be a prime and qq be a power of pp. Frankl first studied pp-modular LL-differencing Sperner systems and showed an upper bound of the form i=0L(ni)\sum_{i=0}^{|L|}\binom{n}{i}. In this paper, we obtain new upper bounds on qq-modular LL-differencing Sperner systems using elementary pp-adic analysis and polynomial method, extending and improving existing results substantially. Moreover, our techniques can be used to derive new upper bounds on subsets of the hypercube with restricted Hamming distances. One highlight of the paper is the first analogue of the celebrated Snevily's theorem in the qq-modular setting, which results in several new upper bounds on qq-modular LL-avoiding LL-intersecting systems. In particular, we improve a result of Felszeghy, Heged\H{u}s, and R\'{o}nyai, and give a partial answer to a question posed by Babai, Frankl, Kutin, and \v{S}tefankovi\v{c}.Comment: 22 pages, results in table 1 and section 6.1 improve

    Bounds on sets with few distances

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    We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of subsets; (2) we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances; (3) we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte. We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.Comment: 11 page
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