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تسجيل مستخدم جديدOn Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs
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For $ngeq s> rgeq 1$ and $kgeq 2$, write $n rightarrow (s)_{k}^r$ if every hyperedge colouring with $k$ colours of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous results by textcite{AGLM14} and textcite{EHMR84} we show that [ text{if } r geq 3 text{ and } n rightarrow (s)_k^r text{ then } 2^n rightarrow (s+1)_{k+3}^{r+1}. ] This yields an improvement for some of the known lower bounds on multicolour hypergraph Ramsey numbers. Given a hypergraph $H=(V,E)$, we consider the Ramsey-like problem of colouring all $r$-subsets of $V$ such that no hyperedge of size $geq r+1$ is monochromatic. We provide upper and lower bounds on the number of colours necessary in terms of the chromatic number $chi(H)$. In particular we show that this number is $O(log^{(r-1)} (r chi(H)) + r)$.
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