Size-Ramsey numbers of graphs with maximum degree three
Nemanja Dragani\'c, Kalina Petrova

TL;DR
This paper improves the upper bound on the size-Ramsey number for graphs with maximum degree three from roughly n^{8/5} to n^{3/2+o(1)} using a novel host graph construction, advancing understanding in Ramsey theory.
Contribution
The paper introduces a new host graph construction to establish a tighter upper bound on size-Ramsey numbers for degree three graphs.
Findings
Bound on size-Ramsey number improved to n^{3/2+o(1)}
Novel host graph construction used instead of binomial random graphs
Results reach a natural barrier of existing methods
Abstract
The size-Ramsey number of a graph is the smallest number of edges a (host) graph can have, such that for any red/blue colouring of , there is a monochromatic copy of in . Recently, Conlon, Nenadov and Truji\'c showed that if is a graph on vertices and maximum degree three, then , improving upon the upper bound of by Kohayakawa, R\"odl, Schacht and Szemer\'edi. In this paper we show that . While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Advanced Topology and Set Theory
