Improved bounds and algorithms for hypergraph 2-coloring
Random Structures & Algorithms
On the chromatic number of set systems
Random Structures & Algorithms
Coloring uniform hypergraphs with few colors
Random Structures & Algorithms
Coloring uniform hypergraphs with few edges
Random Structures & Algorithms
Constructions of sparse uniform hypergraphs with high chromatic number
Random Structures & Algorithms - Special 20th Anniversary Issue
Random Structures & Algorithms - Special 20th Anniversary Issue
An application of Lovász' local lemma‐A new lower bound for the van der Waerden number
Random Structures & Algorithms
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The work deals with a combinatorial problem of P. Erdős and L. Lovász concerning simple hypergraphs. Let $m^{*}(n,r)$ **image** denote the minimum number of edges in an n-uniform simple hypergraph with chromatic number at least $r+1$ **image** . The main result of the work is a new asymptotic lower bound for $m^{*}(n,r)$ **image** . We prove that for large n and r satisfying $r\le n^{1/9}$ **image** the following inequality holds $$m^{*}(n+1,r)\geq{{1}\over{2}}\,r^{2n-2}n^{-6/t},$$ where $t=\left \lfloor\sqrt{\min \left({{\ln n}\over{\ln r}},{{\ln n}\over{2\ln((4/3)\ln n)}}\right)}\right \rfloor$ **image** . This bound improves previously known bounds for $r . The proof is based on a method of random coloring. We have also obtained results concerning colorings of h-simple hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.