An asymmetric random Rado theorem for single equations: The 0‐statement

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 Rodl and Rucinski, and Friedgut, Rodl 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.