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Three‐colored asymmetric bipartite Ramsey number of connected matchings and cycles
Author(s) -
Luo Zhidan,
Peng Yuejian
Publication year - 2020
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22549
Subject(s) - mathematics , ramsey's theorem , combinatorics , bipartite graph , lemma (botany) , monochromatic color , matching (statistics) , colored , ramsey theory , integer (computer science) , discrete mathematics , graph , statistics , physics , computer science , poaceae , optics , biology , programming language , materials science , composite material , ecology
Abstract Guaranteed by Szemerédi's Regularity Lemma, a technique originated by Łuczak is to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component. So determining the Ramsey number of connected matchings is crucial in determining the Ramsey number of cycles. Let k , l , m be integers and r ( k , l , m ) be the minimum integer N such that for any red‐blue‐green coloring of K N , N , there is a red matching of size at least k in a component, or a blue matching of size at least l in a component, or a green matching of size at least m in a component. Bucić, Letzter, and Sudakov determined the exact value of r ( k , l , l ) and led to the asymptotic value of 3‐colored bipartite Ramsey number of cycles (symmetric case). In this paper, we determine the exact value of r ( k , l , m ) completely. This answers a question of Bucić, Letzter, and Sudakov. The crucial part of the proof is the construction we give in Section 4. Applying the technique of Łuczak, we obtain the asymptotic value of 3‐colored bipartite Ramsey number of cycles for all asymmetric cases.