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Register a new userSaturation problems in the Ramsey theory of graphs, posets and point sets
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In 1964, ErdH{o}s, Hajnal and Moon introduced a saturation version of Turans classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the ErdH{o}s-Szekeres theorem about monotone subsequences and saturati
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Given graphs $H_1, dots, H_t$, a graph $G$ is $(H_1, dots, H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a monochromatic $H_i$ in color $i$ for some $iin{1, dots, t}$, but any proper subgraph of $G $ does not possess this property. We define $mathcal{R}_{min}(H_1, dots, H_t)$ to be the family of $(H_1, dots, H_t)$-Ramsey-minimal graphs. A graph $G$ is dfn{$mathcal{R}_{min}(H_1, dots, H_t)$-saturated} if no element of $mathcal{R}_{min}(H_1, dots, H_t)$ is a subgraph of $G$, but for any edge $e$ in $overline{G}$, some element of $mathcal{R}_{min}(H_1, dots, H_t)$ is a subgraph of $G + e$. We define $sat(n, mathcal{R}_{min}(H_1, dots, H_t))$ to be the minimum number of edges over all $mathcal{R}_{min}(H_1, dots, H_t)$-saturated graphs on $n$ vertices. In 1987, Hanson and Toft conjectured that $sat(n, mathcal{R}_{min}(K_{k_1}, dots, K_{k_t}) )= (r - 2)(n - r + 2)+binom{r - 2}{2} $ for $n ge r$, where $r=r(K_{k_1}, dots, K_{k_t})$ is the classical Ramsey number for complete graphs. The first non-trivial case of Hanson and Tofts conjecture for sufficiently large $n$ was setteled in 2011, and is so far the only settled case. Motivated by Hanson and Tofts conjecture, we study the minimum number of edges over all $mathcal{R}_{min}(K_3, mathcal{T}_k)$-saturated graphs on $n$ vertices, where $mathcal{T}_k$ is the family of all trees on $k$ vertices. We show that for $n ge 18$, $sat(n, mathcal{R}_{min}(K_3, mathcal{T}_4)) =lfloor {5n}/{2}rfloor$. For $k ge 5$ and $n ge 2k + (lceil k/2 rceil +1) lceil k/2 rceil -2$, we obtain an asymptotic bound for $sat(n, mathcal{R}_{min}(K_3, mathcal{T}_k))$.
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.
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.
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.
In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,t$, we determine the complementary Ramsey numbers $bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.
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