## Abstract

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 A_{i}x=0, where A_{1},…,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 symmetric random Rado theorem established in a combination of results by Rödl and Ruciński [34] and by Friedgut, Rödl and Schacht [11]. We conjecture that our 1-statement in fact unveils the threshold for the asymmetric random Rado property, yielding a counterpart to the so-called 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, Rödl and Schacht from [11]. 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 [26] for the Kohayakawa-Kreuter conjecture.

Original language | English |
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Article number | 105687 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 193 |

DOIs | |

State | Published - Jan 2023 |

## Keywords

- Container method
- Ramsey theory
- Random graphs and hypergraphs

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics