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1,793 claims from 1,103 papers are on the record. 46 have been checked so far; the other 1,747 have no check with a result yet.

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Keyword: 3-uniform hypergraphs ×Clear all

4 claims from 3 papers

1. Mathematics › Limits and Structures in Graph Theory

   On Ramsey numbers of hedgehogs

   Fox and Li · Combinatorics Probability Computing · 2019

   The paper shows the two-colour Ramsey number of the 3-uniform hedgehog H_t is O(t^2 ln t), answering a question of Conlon, Fox and Rödl about whether it is nearly linear in the hedgehog's vertex count.

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   1. UncheckedThe two-colour Ramsey number of the hedgehog hypergraph H_t is at most of order t squared times the natural log of t.“We answer this question affirmatively, proving that $r(H_t) = O(t^2\ln t)$.”

2. 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 $n$.”

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.”