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We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v ∈ W is(More)
A subset D ⊆ V of a graph G = (V, E) is a (1, j)-set if every vertex v ∈ V \ D is adjacent to at least 1 but not more than j vertices in D. The cardinality of a minimum (1, j)-set of G, denoted as γ (1,j) (G), is called the (1, j)-domination number of G. Given a graph G = (V, E) and an integer k, the decision version of the (1, j)-set problem is to decide(More)
Given a simple directed graph D = (V, A), let the size of the largest induced acyclic tournament be denoted by mat(D). Let D ∈ D(n, p) (with p = p(n)) be a random instance, obtained by randomly orienting each edge of a random graph drawn from G(n, 2p). We show that mat(D) is asymp-totically almost surely (a.a.s.) one of only 2 possible values, namely either(More)
Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least e (1−n −1/12) 1 2 n t ln t(1 2 ln t−1) independent sets. This improves a recent result of the first and third authors [8]. In particular, it implies that as n → ∞, every triangle-free graph on n vertices has at least e (c 1 −o(1)) √ n ln n independent sets,(More)
We obtain new lower bounds for the independence number of K r-free graphs and linear k-uniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tuza [7]. Our proof technique is an extension of a method of Caro and Wei [6, 20], and we also give a new short proof of the main result of [7] using this approach. As(More)
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