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Register a new userAn 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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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.
We introduce an extension, indexed by a partially ordered set P and cardinal numbers k,l, denoted by (k,l)-->P, of the classical relation (k,n,l)--> r in infinite combinatorics. By definition, (k,n,l)--> r holds, if every map from the n-element subsets of k to the subsets of k with less than l elements has a r-element free set. For example, Kuratowskis Free Set Theorem states that (k,n,l)-->n+1 holds iff k is larger than or equal to the n-th cardinal successor l^{+n} of the infinite cardinal k. By using the (k,l)-->P framework, we present a self-contained proof of the first authors result that (l^{+n},n,l)-->n+2, for each infinite cardinal l and each positive integer n, which solves a problem stated in the 1985 monograph of Erdos, Hajnal, Mate, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation (l^{+(n-1)},r,l)-->2^m, where m is the largest integer below (1/2)(1-2^{-r})^{-n/r}, for every infinite cardinal l and all positive integers n and r with r larger than 1 but smaller than n. For example, (aleph_{210},4,aleph_0)-->32,768. Other order-dimension estimates yield relations such as (aleph_{109},4,aleph_0)--> 257 (using an estimate by Furedi and Kahn) and (aleph_7,4,aleph_0)-->10 (using an exact estimate by Dushnik).
We initiate the study of local topology of random graphs. The high level goal is to characterize local motifs in graphs. In this paper, we consider what we call the layer-$r$ subgraphs for an input graph $G = (V,E)$: Specifically, the layer-$r$ subgraph at vertex $u in V$, denoted by $G_{u; r}$, is the induced subgraph of $G$ over vertex set $Delta_{u}^{r}:= left{v in V: d_G(u,v) = r right}$, where $d_G$ is shortest-path distance in $G$. Viewing a graph as a 1-dimensional simplicial complex, we then aim to study the $1$st Betti number of such subgraphs. Our main result is that the $1$st Betti number of layer-$1$ subgraphs in ErdH{o}s--Renyi random graphs $G(n,p)$ satisfies a central limit theorem.
We prove a `resilience version of Diracs theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $varepsilon>0$, a.a.s. the following holds: let $G$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+varepsilon)d$. Then $G$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.
Given $r$-uniform hypergraphs $G$ and $H$ the Turan number $rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the ErdH{o}s-Renyi random $r$-uniform hypergraph, and $H=C_{2ell}^{(r)}$, the $r$-uniform linear cycle of length $2ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $rm ex(G_{n,p}^{(r)}, C_{2ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $ell$ grows. Our main technical contribution is a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the supersaturation result depends on the codegree of some pairs of vertices in the underlying hypergraph. This approach could be used to prove similar results for other hypergraphs $H$.
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