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An asymmetric random Rado theorem for single equations: The 0‐statement
Author(s) -
Hancock Robert,
Treglown Andrew
Publication year - 2022
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21039
Subject(s) - mathematics , combinatorics , partition (number theory) , conjecture , statement (logic) , discrete mathematics , binomial coefficient , binomial (polynomial) , law , statistics , political science
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 ] phaving the asymmetric random Rado property: given partition regular matricesA 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 toA 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 theA 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.