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Status: Unchecked ×Keyword: graph Ramsey theory ×Clear all

5 claims from 3 papers

1. Mathematics › Limits and Structures in Graph Theory

   Ramsey Numbers with Prescribed Rate of Growth

   Pavez‐Signé, Piga and Sanhueza‐Matamala · The Electronic Journal of Combinatorics · 2023

   The note proves that any growth rate between n and the Ramsey number of the complete graph is achieved by connected graphs on n vertices, and that this fails for uniformity 5 and above.

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   1. UncheckedThe paper shows that its result on prescribed Ramsey-number growth rates for graphs has no analogue for hypergraphs of uniformity at least 5.“In contrast, we also show that an analogous statement does not hold for hypergraphs of uniformity at least $5$.”
   2. UncheckedThe authors use their techniques to answer DeBiasio's question on graph sequences with linear 2-colour but superlinear 3-colour Ramsey numbers.“We also use our techniques to answer a question posed by DeBiasio about the existence of sequences of graphs whose $2$-colour Ramsey number is linear whereas their $3$-colour Ramsey number has superlinear growth.”

2. Mathematics › Limits and Structures in Graph Theory

   Upper Bounds for Ordered Ramsey Numbers of Small 1-Orderings

   Kevin · arXiv (Cornell University) · 2017

   The paper makes a first systematic study of ordered Ramsey numbers for graphs with one vertex given a position label, giving upper bounds for small graphs, for paths and for a generalised version.

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   1. UncheckedFor any way of giving one vertex of an n-vertex path a fixed position label, the ordered Ramsey number grows at most linearly in n.“Additionally, for every $1$-ordering $H$ of the $n$-vertex path $P_n$, we prove that $R_<(H) \in O(n)$.”

3. Mathematics › Limits and Structures in Graph Theory

   Induced Ramsey-type theorems

   Fox and Sudakov · arXiv (Cornell University) · 2007

   The paper gives a unified approach to Ramsey-type theorems for graphs with a forbidden induced subgraph, extending earlier results and also showing that pseudo-random graphs have strong induced Ramsey properties.

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   1. UncheckedThe paper's proofs rest on a simple lemma that can stand in for Szemerédi's regularity lemma and, as the authors state, gives much better bounds.“The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds.”
   2. UncheckedThe paper's approach can also be used to show that pseudo-random graphs have strong induced Ramsey properties.“The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties.”