Corpus ID: 221005988

# The minimum degree of minimal Ramsey graphs for cliques.

@article{Bamberg2020TheMD,
title={The minimum degree of minimal Ramsey graphs for cliques.},
author={John Bamberg and Anurag Bishnoi and Thomas Lesgourgues},
journal={arXiv: Combinatorics},
year={2020}
}
• Published 2020
• Mathematics
• arXiv: Combinatorics
We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erdős and Lovasz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of the edges of $G$ contains a monochromatic $K_k$, whereas no proper subgraph of $G$ has this property. The construction used in our proof relies on a group theoretic model of generalised quadrangles introduced by Kantor in 1980.
2 Citations

#### Tables from this paper

On the minimum degree of minimal Ramsey graphs for cliques versus cycles
• Mathematics
• 2021
A graph G is said to be q-Ramsey for a q-tuple of graphs (H1, . . . , Hq), denoted by G →q (H1, . . . , Hq), if every q-edge-coloring of G contains a monochromatic copy of Hi in color i, for some i ∈Expand
Ramsey simplicity of random graphs
• Mathematics
• 2021
A graph G is q-Ramsey for another graph H if in any q-edge-colouring of G there is a monochromatic copy of H, and the classic Ramsey problem asks for the minimum number of vertices in such a graph.Expand

#### References

SHOWING 1-10 OF 29 REFERENCES
Vertex Folkman Numbers and the Minimum Degree of Minimal Ramsey Graphs
• Computer Science, Mathematics
• SIAM J. Discret. Math.
• 2018
The smallest possible minimum degree of $r-color minimal Ramsey graphs for thek-clique is investigated and a bound of the form$O(k^2\log^2 k\big)$is obtained, which is tight up to a$(\log^{2 k)$-factor whenever the number$r\geq2$of colors is fixed. Expand On Ks-free subgraphs in Ks+k-free graphs and vertex Folkman numbers • Mathematics, Computer Science • Comb. • 2011 The upper bound on the function fs,t is improved by showing that fs,s+1(n) ≤ O(n2/3), and it is shown that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n1/2−ɛ) ≤ fs, s+k(n). Expand On a Problem of Formal Logic This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any givenExpand Diagonal Ramsey via effective quasirandomnes • arXiv preprint, • 2020 Diagonal Ramsey via effective quasirandomness We improve the upper bound for diagonal Ramsey numbers to $R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}$ for$k\ge 3$. To do so, we build on a quasirandomness and induction framework for RamseyExpand Packing nearly optimal Ramsey R(3, t) graphs • Mathematics, Computer Science • Comb. • 2020 The algorithmic proof proceeds by sequentially choosing the graphs G_i via a semi-random (i.e., Rodl nibble type) variation of the triangle-free process by approximately decomposing the complete graph K_n into a packing of such nearly optimal Ramsey R(3,t) graphs. Expand A note on Pseudorandom Ramsey graphs • Mathematics, Computer Science • ArXiv • 2019 It is proved that if optimal$K_s$-free pseudorandom graphs exist, then the Ramsey number r(s,t) = t^{s-1+o(1}$ as $t \rightarrow \infty$ as well as improving the exponent of $t$ over the bounds given by the random $F$- free process and random graphs. Expand
Degree Ramsey numbers for even cycles
The main result is that R_\Delta(C_6, s) = \Theta(s^{3/2})$and$R_\ Delta(C_{10, s] = Â£5/4$, which substantially improves the lower bound for$R-Delta (C_{2k}, s)$for general$k\$. Expand
On the minimum degree of minimal Ramsey graphs for multiple colours
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 2016
The dependency of s r ( K k ) on r is studied and it is shown that, under the condition that k is constant, s r( K k) = r 2 ź polylog r . Expand
Recent developments in graph Ramsey theory
• Mathematics, Computer Science
• Surveys in Combinatorics
• 2015
There has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. Expand