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arXiv · 2004.14076

An asymmetric random Rado theorem for single equations: the $0$-statement

Robert Hancock·Andrew Treglown
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Abstract

A famous result of Rado characterises those integer matrices $A$ which are partition regular, i.e. for which any finite colouring of the positive integers gives rise to a monochromatic solution to the equation $Ax=0$. Aigner-Horev and Person recently stated a conjecture on the probability threshold for the binomial random set $[n]_p$ having the asymmetric random Rado property: given partition regular matrices $A_1, \dots, A_r$ (for a fixed $r \geq 2$), however one $r$-colours $[n]_p$, there is always a colour $i \in [r]$ such that there is an $i$-coloured solution to $A_i x=0$. This generalises the symmetric case, which was resolved by R\"odl and Ruci\'nski, and Friedgut, R\"odl and Schacht. Aigner-Horev and Person proved the $1$-statement of their asymmetric conjecture. In this paper, we resolve the $0$-statement in the case where the $A_i x=0$ correspond to single linear equations. Additionally we close a gap in the original proof of the 0-statement of the (symmetric) random Rado theorem.

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Robert Hancock, Andrew Treglown. 2020-04-29. An asymmetric random Rado theorem for single equations: the $0$-statement. https://arxiv.org/abs/2004.14076

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