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Keyword: hypergraph Ramsey numbers ×Clear all

7 claims from 4 papers

1. Mathematics › Limits and Structures in Graph Theory

   Set-coloring Ramsey numbers via codes

   Conlon, Fox, He, Mubayi, Suk and Verstraëte · arXiv (Cornell University) · 2022

   The paper proves general upper and lower bounds on set-coloring Ramsey numbers, using a link to error-correcting codes, and also studies the analogous problem for hypergraphs.

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   1. UncheckedThe set-coloring Ramsey number R(n;r,s) is shown to grow as 2 to the power of a constant multiple of nr when s/r stays away from 0 and 1.“We prove general upper and lower bounds on $R(n;r,s)$ which imply that $R(n;r,s) = 2^{Θ(nr)}$ if $s/r$ is bounded away from $0$ and $1$.”

2. Mathematics › Limits and Structures in Graph Theory

   Hypergraph Ramsey numbers

   Conlon, Fox and Sudakov · arXiv (Cornell University) · 2008

   The paper gives new upper and lower bounds for several hypergraph Ramsey numbers, including r_3(s,n) and the three-colour number r_3(n,n,n), and makes progress on related Ramsey-type problems.

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   1. UncheckedFor colourings of triples, the paper gives a new upper bound on r_3(s,n) of 2^(n^(s-2) log n), with s fixed, improving the exponent of Erdős and Rado's 1952 bound.“In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952.”
   2. Unchecked“We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq 2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n.”

3. Mathematics › Limits and Structures in Graph Theory

   On off-diagonal hypergraph Ramsey numbers

   David, Jacob, Benjamin et al. · arXiv (Cornell University) · 2024

   The paper studies how fast off-diagonal hypergraph Ramsey numbers grow, giving a broad lower bound of 2^{Ω(n log n)} and a linear hypergraph whose Ramsey number is superpolynomial in n.

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   1. UncheckedFor a broad class of 3-uniform hypergraphs H, including links of odd cycles and some tight cycles, the Ramsey number r(H, K_n) is at least 2^{Ω(n log n)}.“First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{Ω_H(n \log n)}$.”
   2. UncheckedThe paper shows a linear 3-uniform hypergraph H exists whose Ramsey number against the complete hypergraph on n vertices grows faster than any polynomial in n.“Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$.”

4. Mathematics › Limits and Structures in Graph Theory

   Ramsey numbers of hypergraphs of a given size

   Bradač, Fox and Sudakov · arXiv (Cornell University) · 2023

   The paper bounds how large the Ramsey number of a k-uniform hypergraph with m edges can be, and gives a construction showing the bound is tight for four or more colours.

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   1. UncheckedFor fixed uniformity k≥3 and q≥2 colours, hypergraphs with m edges have q-colour Ramsey number at most a k-level tower of height O(√m).“For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function.”
   2. UncheckedThe paper presents a construction showing its upper bound on hypergraph Ramsey numbers by edge count is tight when there are at least four colours.“We also present a construction showing that this bound is tight for $q \ge 4$.”