Large stars with few colors
Amir Khamseh, Gholam Reza Omidi

TL;DR
This paper investigates a generalized Ramsey problem focusing on covering vertices with limited monochromatic colors in edge-colored complete graphs, specifically for stars and near-complete color coverage cases.
Contribution
It provides new bounds and results for the minimal number of vertices needed to guarantee a star with at most s colors in t-colored complete graphs, extending previous work.
Findings
Established bounds for R_{s,t}(G) when G is a star and s=t-1 or s=t-2
Generalized Burr and Roberts' results to broader color coverage scenarios
Connected the problem to classical Ramsey theory and monochromatic covering concepts.
Abstract
A recent question in generalized Ramsey theory is that for fixed positive integers , at least how many vertices can be covered by the vertices of no more than monochromatic members of the family in every edge coloring of with colors. This is related to an old problem of Chung and Liu: for graph and integers what is the smallest positive integer such that every coloring of the edges of with colors contains a copy of with at most colors. We answer this question when is a star and is either or generalizing the well-known result of Burr and Roberts.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
