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

Abstract

A famous result of Rado characterizes those integer matrices A which are partition regular, that is, for which any finite coloring 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 , … , A r (for a fixed r ≥ 2), however one r-colors [n] p , there is always a color i ∈ [r] such that there is an i-colored solution to A i x = 0. This generalizes 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.