An asymmetric random Rado theorem: 1-statement

A classical result by Rado characterises the so-called partition-regular matrices A, i.e. those matrices A for which any finite colouring of the positive integers yields a monochromatic solution to the equation Ax=0. We study the asymmetric random Rado problem for the (binomial) random set [n]p in which one seeks to determine the threshold for the property that any r-colouring, r≥2, of the random set has a colour i∈[r] admitting a solution for the matrical equation Aix=0, where A1,…,Ar are predetermined partition-regular matrices pre-assigned to the colours involved.