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arXiv · 2606.02873

A Sparse Transference Principle for a Non-Monotone Ramsey Property

Gaia Carenini
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Abstract

We prove a sparse transference theorem for induced Ramsey graphs. The theorem transfers the weighted random-host proof of Arag\~ao, Campos, Dahia, Filipe, and Marciano to the sparse random setting. It follows that, for every fixed graph $H$ with no isolated vertices and at least two edges, and every $\eta>0$, there is $C>0$ such that, whenever $N\ge r^{Cr}$ and $N^{-1/m_2(H)+\eta}\le p\le \frac12$, with high probability every $r$-colouring of the edges of $G(N,p)$ contains a monochromatic induced copy of $H$. Here, $m_2(H)$ denotes the usual maximum 2-density of $H$.

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Gaia Carenini. 2026-06-01. A Sparse Transference Principle for a Non-Monotone Ramsey Property. https://arxiv.org/abs/2606.02873

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