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 $r\ge 2$), 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 Rödl and Ruciński, and Friedgut, Rödl 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.

arXiv comment20 pages, 4 figures, author accepted version, to appear in Random Structures and Algorithms