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arXiv · 1409.6921

Improved algorithms for colorings of simple hypergraphs and applications

Jakub Kozik·Dmitry Shabanov
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Abstract

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any $n$-uniform simple (i.e. every two distinct edges share at most one vertex) hypergraph $H$ with maximum edge degree at most \[ \Delta(H)\leq c\cdot nr^{n-1}, \] is $r$-colorable, where $c>0$ is an absolute constant. %We prove also that similar result holds for $b$-simple hypergraphs. As an application of our proof technique we establish a new lower bound for Van der Waerden number $W(n,r)$, the minimum $N$ such that in any $r$-coloring of the set $\{1,...,N\}$ there exists a monochromatic arithmetic progression of length $n$. We show that \[ W(n,r)>c\cdot r^{n-1}, \] for some absolute constant $c>0$.

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Jakub Kozik, Dmitry Shabanov. 2014-09-24. Improved algorithms for colorings of simple hypergraphs and applications. https://arxiv.org/abs/1409.6921

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