169 research outputs found

    Cross-intersecting families of vectors

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    Given a sequence of positive integers p=(p1,...,pn)p = (p_1, . . ., p_n), let SpS_p denote the family of all sequences of positive integers x=(x1,...,xn)x = (x_1,...,x_n) such that xipix_i \le p_i for all ii. Two families of sequences (or vectors), A,BSpA,B \subseteq S_p, are said to be rr-cross-intersecting if no matter how we select xAx \in A and yBy \in B, there are at least rr distinct indices ii such that xi=yix_i = y_i. We determine the maximum value of AB|A|\cdot|B| over all pairs of rr- cross-intersecting families and characterize the extremal pairs for r1r \ge 1, provided that minpi>r+1\min p_i >r+1. The case minpir+1\min p_i \le r+1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang

    A constructive proof of the general Lovasz Local Lemma

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    The Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its restrictions were subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06, Sri08, Mos08]. In [Mos09], a constructive proof was presented that works under negligible restrictions, formulated in terms of the Bounded Occurrence Satisfiability problem. In the present paper, we reformulate and improve upon these findings so as to directly apply to almost all known applications of the general Local Lemma.Comment: 8 page

    Conflict-free coloring of graphs

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    We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page
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