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تسجيل مستخدم جديدAn Asymmetric Random Rado Theorem: 1-statement
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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 {sl 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, $r geq 2$, of the random set has a colour $i in [r]$ admitting a solution for the matrical equation $A_i x = 0$, where $A_1,ldots,A_r$ are predetermined partition-regular matrices pre-assigned to the colours involved. We prove a $1$-statement for the asymmetric random Rado property. In the symmetric setting our result retrieves the $1$-statement of the {sl symmetric} random Rado theorem established in a combination of results by Rodl and Rucinski~cite{RR97} and by Friedgut, Rodl and Schacht~cite{FRS10}. We conjecture that our $1$-statement in fact unveils the threshold for the asymmetric random Rado property, yielding a counterpart to the so-called {em Kohayakawa-Kreuter conjecture} concerning the threshold for the asymmetric random Ramsey problem in graphs. We deduce the aforementioned $1$-statement for the asymmetric random Rado property after establishing a broader result generalising the main theorem of Friedgut, Rodl and Schacht from~cite{FRS10}. The latter then serves as a combinatorial framework through which $1$-statements for Ramsey-type problems in random sets and (hyper)graphs alike can be established in the asymmetric setting following a relatively short combinatorial examination of certain hypergraphs. To establish this framework we utilise a recent approach put forth by Mousset, Nenadov and Samotij~cite{MNS18} for the Kohayakawa-Kreuter conjecture.
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