Ramsey for complete graphs with dropped cliques
Jonathan Chappelon, Luis Pedro Montejano, Jorge Luis Ram\'irez, Alfons\'in

TL;DR
This paper establishes two explicit upper bounds for Ramsey numbers involving graphs formed by dropping cliques from complete graphs, introducing a new coloring method and discussing potential improvements over classical bounds.
Contribution
The paper provides two new upper bounds for Ramsey numbers of graphs with dropped cliques, including a novel coloring approach called $ ext{ extbackslash chi}_r$-colorings.
Findings
First upper bound generalizes classical bounds for uniform cases.
Second upper bound uses $ ext{ extbackslash chi}_r$-colorings to potentially improve classical bounds.
Discussion of a conjecture suggesting the second bound is often tighter.
Abstract
Let be the complete graph on vertices from which a set of edges, induced by a clique of order , has been dropped. In this note we give two explicit upper bounds for (the smallest integer such that for any -edge coloring of there always occurs a monochromatic for some ). Our first upper bound contains a classical one in the case when and for all . The second one is obtained by introducing a new edge coloring called {\em -colorings}. We finally discuss a conjecture claiming, in particular, that our second upper bound improves the classical one in infinitely many cases.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory
