Ramsey goodness of books revisited
Jacob Fox, Xiaoyu He, Yuval Wigderson

TL;DR
This paper provides a new, simpler proof that certain book graphs are Ramsey good for large enough parameters, avoiding complex regularity lemmas and establishing bounds on the size of graphs involved.
Contribution
It introduces a short, regularity-free proof for the Ramsey goodness of book graphs and extends results to more general sparse graphs with explicit bounds.
Findings
Every $B_{k,n}$ with $n \\geq 2^{k^{10p}}$ is $p$-good.
New bounds on Ramsey numbers for specific sparse graphs without using regularity lemmas.
Extended the class of graphs for which tight Ramsey bounds are known.
Abstract
The Ramsey number is the minimum such that every graph on vertices contains as a subgraph or its complement contains as a subgraph. For integers , the -book is the graph on vertices consisting of a copy of , called the spine, as well as additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph on vertices is called -good if . Nikiforov and Rousseau proved that if is sufficiently large in terms of and , then is -good. Their proof uses Szemer\'edi's regularity lemma and gives a tower-type bound on . We give a short new proof that avoids using the regularity method and shows that every with is -good. Using Szemer\'edi's regularity lemma, Nikiforov and Rousseau also proved…
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