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Colourings of Uniform Hypergraphs with Large Girth and Applications

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  09 October 2017

ANDREY KUPAVSKII  and
DMITRY SHABANOV
ANDREY KUPAVSKII
Affiliation:
Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics and Technology, Moscow Region, Russia Ecole Polytechnique Fédérale de Lausanne, Switzerland (e-mail: kupavskii@ya.ru)
DMITRY SHABANOV
Affiliation:
Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics and Technology, 141700, Institutskiy per. 9, Dolgoprudny, Moscow Region, Russia Faculty of Computer Science, National Research University Higher School of Economics (HSE), 101000, Myasnitskaya Str. 20, Moscow, Russia (e-mail: dm.shabanov.msu@gmail.com)
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Abstract

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that if H is an n-uniform non-r-colourable simple hypergraph then its maximum edge degree Δ(H) satisfies the inequality

$$ \Delta(H)\geqslant c\cdot r^{n-1}\ffrac{n(\ln\ln n)^2}{\ln n} $$
for some absolute constant c > 0.

As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden number W(n, r), the minimum natural N such that in an arbitrary colouring of the set of integers {1,. . .,N} with r colours there exists a monochromatic arithmetic progression of length n. We prove that

$$ W(n,r)\geqslant c\cdot r^{n-1}\ffrac{(\ln\ln n)^2}{\ln n}. $$

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05D40: Probabilistic methods 05D10: Ramsey theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 245 - 273
Copyright
Copyright © Cambridge University Press 2017 

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