Ramsey-type numbers involving graphs and hypergraphs with large girth
H. H\`an, T. Retter, V. R\"odl, M. Schacht

TL;DR
This paper investigates the minimal size of graphs with large girth that guarantee monochromatic cycles under any edge coloring, connecting Ramsey numbers, graph girth, and related combinatorial problems.
Contribution
It establishes upper bounds on the size of such graphs based on Ramsey numbers and girth, advancing understanding of Ramsey-type problems in graphs and hypergraphs.
Findings
Existence of graphs with specified girth and Ramsey properties on bounded vertices
Derived bounds involving Ramsey numbers and girth for minimal graphs
Explored related problems in arithmetic progressions and clique formations
Abstract
A question of Erd\H{o}s asks if for every pair of positive integers and , there exists a graph having and the property that every -colouring of the edges of yields a monochromatic cycle . The existence of such graphs was confirmed by the third author and Ruci\'nski. We consider the related numerical problem of determining the smallest such graph with this property. We show that for integers and , there exists a graph on vertices (where is the -colour Ramsey number for the cycle ) having and the Ramsey property that every -colouring of yields a monochromatic . Two related numerical problems regarding arithmetic progressions in sets and cliques in graphs are also considered.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
