An asymmetric random Rado theorem for single equations: The 0‐statement. Issue 4 (21st August 2021) - Hancock, Robert; Treglown, Andrew
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 A x = 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.