IA Scholar Query: Infinite Monochromatic Paths and a Theorem of Erdős-Hajnal-Rado.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 02 Jun 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440The Ramsey Theory of Henson graphs
https://scholar.archive.org/work/sgiq2irzmvamvlvpa2ltan7kyy
Analogues of Ramsey's 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 author's recent result for the triangle-free Henson graph, we prove that for each k≥ 4, the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey's 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.Natasha Dobrinenwork_sgiq2irzmvamvlvpa2ltan7kyyThu, 02 Jun 2022 00:00:00 GMTOn quantitative aspects of a canonisation theorem for edge-orderings
https://scholar.archive.org/work/c4riaqr3are73cmprt2gmlvroa
For integers k≥ 2 and N≥ 2k+1 there are k!2^k canonical orderings of the edges of the complete k-uniform hypergraph with vertex set [N] = {1,2,..., N}. These are exactly the orderings with the property that any two subsets A, B⊆ [N] of the same size induce isomorphic suborderings. We study the associated canonisation problem to estimate, given k and n, the least integer N such that no matter how the k-subsets of [N] are ordered there always exists an n-element set X⊆ [N] whose k-subsets are ordered canonically. For fixed k we prove lower and upper bounds on these numbers that are k times iterated exponential in a polynomial of n.Christian Reiher, Vojtěch Rödl, Marcelo Sales, Kevin Sames, Mathias Schachtwork_c4riaqr3are73cmprt2gmlvroaThu, 12 May 2022 00:00:00 GMTNew Stepping-Up Constructions for Multicoloured Hypergraphs
https://scholar.archive.org/work/h7g76fsjkfenvnv3qcwpgqgmcu
Generalizing the classical Ramsey numbers, r_k(t; q, p) is the smallest integer n such that every q-colouring of the k-sets on n vertices contains a set of t vertices spanning fewer than p colours. We prove the first tower-type lower bounds on these numbers via two new stepping-up constructions, both variants of the original stepping-up lemma due to Erdős and Hajnal. We use these to resolve a problem of Conlon, Fox, and Rödl. More precisely, we construct a family of hypergraphs with arbitrarily large tower height separation between their 2-colour and q-colour Ramsey numbers.Quentin Dubroff, António Girão, Eoin Hurley, Corrine Yapwork_h7g76fsjkfenvnv3qcwpgqgmcuMon, 28 Feb 2022 00:00:00 GMTGeneralizations and Strengthenings of Ryser's Conjecture
https://scholar.archive.org/work/wscxvkvazffy5hc4hferceekku
Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem is only known to be true for $r\leq 3$, or when $\nu(H)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $\alpha(G)$, there exists at most $(r-1)\alpha(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.Louis DeBiasio, Yigal Kamel, Grace McCourt, Hannah Sheatswork_wscxvkvazffy5hc4hferceekkuFri, 03 Dec 2021 00:00:00 GMTGeneralizations and strengthenings of Ryser's conjecture
https://scholar.archive.org/work/rmaa6toeo5f7xeq2bcraydzfhy
Ryser's conjecture says that for every r-partite hypergraph H with matching number ν(H), the vertex cover number is at most (r-1)ν(H). This far reaching generalization of König's theorem is only known to be true for r≤ 3, or ν(G)=1 and r≤ 5. An equivalent formulation of Ryser's conjecture is that in every r-edge coloring of a graph G with independence number α(G), there exists at most (r-1)α(G) monochromatic connected subgraphs which cover the vertex set of G. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.Louis DeBiasio, Yigal Kamel, Grace McCourt, Hannah Sheatswork_rmaa6toeo5f7xeq2bcraydzfhyWed, 03 Nov 2021 00:00:00 GMT