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A famous result of Rado characterises those integer matrices $A$ which arepartition regular, i.e. for which any finite colouring of the positive integersgives rise to a monochromatic solution to the equation $Ax=0$. Aigner-Horev andPerson recently stated a conjecture on the probability threshold for thebinomial random set $[n]_p$ having the asymmetric random Rado property: givenpartition regular matrices $A_1, \dots, A_r$ (for a fixed $r \geq 2$), howeverone $r$-colours $[n]_p$, there is always a colour $i \in [r]$ such that thereis an $i$-coloured solution to $A_i x=0$. This generalises the symmetric case,which was resolved by R\"odl and Ruci\'nski, and Friedgut, R\"odl and Schacht.Aigner-Horev and Person proved the $1$-statement of their asymmetricconjecture. 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 inthe original proof of the 0-statement of the (symmetric) random Rado theorem.

An asymmetric random Rado theorem for single equations: the $0$-statement