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| Év szerint | Hónap szerint | Ugrás a hónaphoz | |||||||
Andrea Freschi (University of Birmingham): Typical Ramsey properties of abelian groups |
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| A classical result of Rado characterises all those integer matrices $A$ for which any finite colouring of $\mathbb N$ yields a monochromatic solution to the system of equations $Ax=0$. Rödl and Rucinski \cite{rr} and Friedgut, Rödl and Schacht \cite{frs} proved a random version of Rado’s theorem where one considers a random subset of the first $n$ positive integers $\{1,2,\dots,n\}$ instead of $\mathbb N$. In this talk, we consider the analogous random Ramsey problem in the more general setting of abelian groups. Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r\in\mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. Given a well-behaved sequence $(S_n)_{n\in\mathbb N}$ of $(A,r)$-Rado finite subsets of abelian groups, we are interested in determining the probability threshold $\hat p:=\hat p(n)$ such that $$\lim _{n \rightarrow \infty} \mathbb P [ S_{n,p} \text{ is } (A,r)\text{-Rado}]= \begin{cases} 0 &\text{ if } p=o(\hat p); \\ 1 &\text{ if } p=\omega(\hat p). \end{cases}$$ where $S_{n,p}$ denotes the random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. Our main result is a general black box to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. For example, a consequence of the Green--Tao theorem \cite{gt} is the \emph{van der Waerden theorem for the primes}: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $S_n:=[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem. This talk is based on joint work with Robert Hancock and Andrew Treglown. |
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