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Register a new userBorel sets of Rado graphs and Ramseys Theorem
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Natasha Dobrinen
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The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset $mathcal{X}subseteq [omega]^{omega}$, where $[omega]^{omega}$ is endowed with the metric topology, each infinite subset $Xsubseteq omega$ contains an infinite subset $Ysubseteq X$ such that $[Y]^{omega}$ is either contained in $mathcal{X}$ or disjoint from $mathcal{X}$. Kechris, Pestov, and Todorcevic point out in their seminal 2005 paper the dearth of similar results for homogeneous structures. Such results are a necessary step to the larger goal of finding a correspondence between structures with infinite dimensional Ramsey properties and topological dynamics, extending their correspondence between the Ramsey property and extreme amenability. In this article, we prove an analogue of the Galvin-Prikry theorem for the Rado graph. Any such infinite dimensional Ramsey theorem is subject to constraints following from the 2006 work of Laflamme, Sauer, and Vuksanovic. The proof uses techniques developed for the authors work on the Ramsey theory of the Henson graphs as well as some new methods for fusion sequences, used to bypass the lack of a certain amalgamation property enjoyed by the Baire space.
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We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Bore
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.
Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number $mathfrak v = mathfrak v(n)$ such that, if $A_1, ldots, A_n$ are $n$ equivalence relations on a common finite ground set $X$, such that for each $i$ there are at least $mathfrak v$ elements of $X$ that belong to $A_i$-equivalence classes of size larger than $1$, then $X$ has a rainbow matching---a set of $2n$ distinct elements $a_1, b_1, ldots, a_n, b_n$, such that $a_i$ is $A_i$-equivalent to $b_i$ for each $i$? Grinblat has shown that $mathfrak v(n) le 10n/3 + O(sqrt{n})$. He asks whether $mathfrak v(n) = 3n-2$ for all $nge 4$. In this paper we improve the upper bound (for all large enough $n$) to $mathfrak v(n) le 16n/5 + O(1)$.
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $sgeq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erdos--Renyi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measures on the space of such graphs. The main idea of our construction is based on the new notions of {it measurable universal}, and {it topologically universal} graphs, which are interesting themselves. The realization of the construction can be regarded as two-step randomization for universal measurable graph : {it randomization in vertices} and {it randomization in edges}. For $K_s$-free, $sgeq 3$ there is only randomization in vertices of the measurable graphs. The completeness of our lists is proved using the important theorem by D. Aldous about $S_{infty}$-invariant matrices, which we reformulate in appropriate way.
We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Millikens theorem on trees.
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