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arXiv · 1910.00136

Vertex Ramsey properties of randomly perturbed graphs

Shagnik Das·Patrick Morris·Andrew Treglown
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Abstract

Given graphs $F,H$ and $G$, we say that $G$ is $(F,H)_v$-Ramsey if every red/blue vertex colouring of $G$ containsa red copy of $F$ or a blue copy of $H$. Results of {\L}uczak, Ruci\'nski and Voigt, and Kreuter determine the threshold for the property that the random graph $G(n,p)$ is $(F,H)_v$-Ramsey. In this paper we consider the sister problem in the setting of \emph{randomly perturbed graphs}. In particular, we determine how many random edges one needs to add to a dense graph to ensure that with high probability the resulting graph is $(F,H)_v$-Ramsey for all pairs $(F,H)$ that involve at least one clique.

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Shagnik Das, Patrick Morris, Andrew Treglown. 2019-09-30. Vertex Ramsey properties of randomly perturbed graphs. https://arxiv.org/abs/1910.00136

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