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Register a new userA natural generalisation in graph Ramsey theory
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In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1le rle e(H)$ such graphs occur naturally at intermediate steps in the synthesis of a $2$-colour Ramsey graph $G_1longrightarrow H$. (The corresponding notion of Ramsey-type numbers was introduced by Erdos, Hajnal and Rado in 1965 and subsequently studied by Erdos and Szemeredi in 1972). For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and establish Ramsey-infiniteness. From the structural point of view, we characterise the class of the minimal $G_r$ in the case when $H$ is relaxed to be the graph property of containing a cycle; we then use it to progress towards a constructive description of that class by proving both a reduction and an extension theorem.
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Building on previous work of the author, for each finite triangle-free graph $mathbf{G}$, we determine the equivalence relation on the copies of $mathbf{G}$ inside the universal homogeneous triangle-free graph, $mathcal{H}_3$, with the smallest number of equivalence classes so that each one of the classes persists in every isomorphic subcopy of $mathcal{H}_3$. This characterizes the exact big Ramsey degrees of $mathcal{H}_3$. It follows that the triangle-free Henson graph is a big Ramsey structure.
For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $rin mathbb{N}$, the $r$-independence number of $G$, denoted $alpha_r(G)$ is the largest size of a $K_r$-free set of vertices in $G$. In this paper, we discuss Ramsey--Turan-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. For cliques, we show that for any $kgeq 3$ and $eta>0$, any graph $G$ on $n$ vertices with $delta(G)geq eta n$ and $alpha_k(G)=o(n)$ has a $K_k$-tiling covering all but $lfloortfrac{1}{eta}rfloor(k-1)$ vertices. All conditions in this result are tight; the number of vertices left uncovered can not be improved and for $eta<tfrac{1}{k}$, a condition of $alpha_{k-1}(G)=o(n)$ would not suffice. When $eta>tfrac{1}{k}$, we then show that $alpha_{k-1}(G)=o(n)$ does suffice, but not $alpha_{k-2}(G)=o(n)$. These results unify and generalise previous results of Balogh-Molla-Sharifzadeh, Nenadov-Pehova and Balogh-McDowell-Molla-Mycroft on the subject. We further explore the picture when $F$ is a tree or a cycle and discuss the effect of replacing the independence number condition with $alpha^*(G)=o(n)$ (meaning that any pair of disjoint linear sized sets induce an edge between them) where one can force perfect $F$-tilings covering all the vertices. Finally we discuss the consequences of these results in the randomly perturbed setting.
Analogues of Ramseys Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the authors recent result for the triangle-free Henson graph, we prove that for each $kge 4$, the $k$-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramseys Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker.
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerdens theorem and Szemeredis theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. $bullet$ We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. $bullet$ We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in $mathbb{Z}$, $mathbb{Z}^d$, and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. $bullet$ We show that, for any countable abelian group $G$, any F{o}lner sequence $Phi$ in $G$, and any $c in (0, 1)$, there exists a discordant set $A subseteq G$ with $d_Phi(A) = c$. Here $d_Phi$ denotes density along $Phi$. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.
Combining two classical notions in extremal combinatorics, the study of Ramsey-Turan theory seeks to determine, for integers $mle n$ and $p leq q$, the number $mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$. Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollobas-ErdH{o}s graphs with densities other than $1/2$. We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollobas-ErdH{o}s-type graphs using high dimensional complex spheres with all rational densities. Some matching upper bounds are also provided.
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