Some new bounds on partition critical hypergraphs

Zoltán Füredi, Attila Sali

Research output: Contribution to journalArticle

Abstract

A hypergraph ([n],E) is 3-color critical if it is not 2-colorable, but for all E∈E the hypergraph ([n],E\{E}) is 2-colorable. Lovász proved in 1976, that {pipe}E{pipe}≤nk-1 if E is k-uniform. Here we give a new algebraic proof and an ordered version that is a sharpening of Lovász' result.Let E⊆[n]k be a k-uniform set system on an underlying set [n] of n elements. Let us fix an ordering E 1, E 2, . . .E t of E and a prescribed partition {A i, B i} of each E i (i.e., A i∪B i=E i and A i∩B i=∅). Assume that for all i=1, 2, . . ., t there exists a partition {C i, D i} of [n] such that E i∩C i=A i and E i∩D i=B i, but {E j∩C i, E j∩D i}≠{A j, B j} for all j<i. That is, the ith partition cuts the ith set as it was prescribed, but it does not cut any earlier set properly. Then t≤f(n,k):=n-1k-1+n-1k-2+⋯+n-10. This is sharp for k=2, 3. We show that this upper bound is almost the best possible, at least the first three terms are correct; we give constructions of size f(n, k)-O(n k-4) (for k fixed and n→∞). We also give constructions of sizes nk-1 for all n and k.Furthermore, in the 3-color-critical case (i.e. {A i, B i}={E i, ∅} for all i), t≤nk-1.

Original languageEnglish
Pages (from-to)844-852
Number of pages9
JournalEuropean Journal of Combinatorics
Volume33
Issue number5
DOIs
Publication statusPublished - Jul 1 2012

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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