3‐Color bipartite Ramsey number of cycles and paths

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 at least size l in a component, or a green matching of size at least m in a component. In this paper, we determine the exact value of r(k, l, m) completely. Applying a technique originated by Luczak that applies Szemerédi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain the 3-colored asymmetric bipartite Ramsey number of cycles asymptotically.

Section: ⎧ ⎨ ⎩mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

“…The proof follows from [3]. Let G be a graph as in Theorem A.1 and edges of G are colored by red, blue, and green.…”

Section: A1 | Monochromatic Connected Matchings In Almost Complete Bmentioning

confidence: 99%

“…Note that

r (1, l, m) = r (l, m)

. For the remaining cases, we apply König's theorem as in [3], and determine the exact value of

r (k, l, m)

completely as stated in the following theorem. The key of the success of obtaining Theorem 1.2 is the construction given in Lemma 4.1.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we give some useful facts. In Section 3, we give the proof of

r (k, l) = k + l - 1

which mentioned in [3] without a detailed proof. In Section 4, we give the proof of Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Three‐colored asymmetric bipartite Ramsey number of connected matchings and cycles

Luo

Peng

2020

Monochromatic connected matchings in 2‐edge‐colored multipartite graphs

Balogh

Kostochka

Lavrov

et al. 2022

A matching M $M$ in a graph G $G$ is connected if all the edges of M $M$ are in the same component of G $G$ . Following Łuczak, there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey‐type bounds on the sizes of monochromatic connected matchings in 2‐edge‐colored multipartite graphs. In addition, we prove a stability theorem for such matchings.

The (t−1) $(t-1)$‐chromatic Ramsey number for paths

Bucić

Khamseh

2022

Self Cite

The following relaxation of the classical problem of determining the Ramsey number of a fixed graph has first been proposed by Erdős, Hajnal and Rado over 50 years ago. Given a graph G $G$ and an integer t ≥ 2 $t\ge 2$ determine the minimum number N $N$ such that in any t $t$‐coloured complete graph on N $N$ vertices there is a copy of G $G$ using only edges of some t − 1 $t-1$ colours. We determine the answer precisely when G $G$ is a path.