Improved algorithms for colorings of simple hypergraphs and applications

TL;DR

This paper introduces improved algorithms for coloring simple hypergraphs using a random recoloring method, establishing new bounds on hypergraph colorability and Van der Waerden numbers.

Contribution

It presents a novel random recoloring algorithm that proves hypergraphs with certain degree bounds are r-colorable and derives new lower bounds for Van der Waerden numbers.

Findings

Any n-uniform simple hypergraph with degree ≤ c·n·r^{n-1} is r-colorable.

Established a new lower bound W(n,r) > c·r^{n-1} for Van der Waerden numbers.

The algorithm advances understanding of hypergraph colorings and arithmetic progression bounds.

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 $c>0$.