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1,860 claims from 1,147 papers are on the record. 46 have been checked so far; the other 1,814 have no check with a result yet.

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Keyword: k-uniform hypergraphs ×Clear all

4 claims from 3 papers

1. Mathematics › Limits and Structures in Graph Theory

   Ramsey numbers of sparse hypergraphs

   Conlon, Fox and Sudakov · arXiv (Cornell University) · 2007

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   1. Unchecked“Moreover, our method demonstrates that, for k \geq 4, c(Δ, k) \leq 2^{2^{\Ddots^{2^{c Δ}}}}, where the tower is of height k and the constant c depends on k.”

2. 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 $q \ge 4$.”

3. Mathematics › Limits and Structures in Graph Theory

   Ramsey goodness of $k$-uniform paths, or the lack thereof

   Boyadzhiyska and Lo · arXiv (Cornell University) · 2023

   The paper explores Ramsey goodness for k-uniform hypergraph paths, showing that unlike graph paths, many are not H-good, while long loose paths are asymptotically good and tight paths are settled for some 3-graphs.

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   1. UncheckedLong loose paths in k-uniform hypergraphs are always at least asymptotically H-good for any hypergraph H, with bounds the paper says are best possible in a certain sense.“On the other hand, we prove that long loose paths are always at least asymptotically $H$-good for every $H$ and derive lower and upper bounds that are best possible in a certain sense.”