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Colourings of Uniform Hypergraphs with Large Girth and Applications

  • ANDREY KUPAVSKII (a1) (a2) and DMITRY SHABANOV (a3) (a4)

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

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

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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