For the given bipartite graphs G 1 , . . . , G n , the bipartite Ramsey number BR ( G 1 , . . . , G n ) is the least positive integer b such that any complete bipartite graph K b,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . For the given bipartite graphs G 1 , . . . , G n and a positive integer m , the m -bipartite Ramsey number BR m ( G 1 , . . . , G n ) is deﬁned as the least positive integer b ( b ≥ m ) such that any complete bipartite graph K m,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . The values of BR m ( G 1 , G 2 ) (for each m ), BR m ( K 3 , 3 , K 3 , 3 ) and BR m ( K 2 , 2 , K 5 , 5 ) (for particular values of m ) have already been determined in several articles, where G 1 = K 2 , 2 and G 2 ∈ { K 3 , 3 , K 4 , 4 } . In this article, the value of BR m ( K 2 , 2 , K 6 , 6 ) is computed for each m ∈ { 2 , 3 , . . . , 8 } .

/pdf/the-m-bipartite-ramsey-number-of-the-k-22-versus-k-66-2m0qxadf.pdf

The m-Bipartite Ramsey Number of the K{2,2} Versus K{6,6}

Star-critical Ramsey numbers involving large books

This book is a tribute to Paul Erd\H{o}s, the wandering mathematician once described as the "prince of problem solvers and the absolute monarch of problem posers." It examines -- within the context of his unique personality and lifestyle -- the legacy of open problems he left to the world after his death in 1996. Unwilling to succumb to the temptations of money and position, Erd\H{o}s never had a home and never held a job. His "home" was a bag or two containing all his belongings and a record of the collective activities of the mathematical community. His "job" was one at which he excelled: identifying a fundamental roadblock in some particular line of approach and capturing it in a well-chosen, often innocent-looking problem, whose solution would likewise provide insight into the underlying theory. By cataloguing the unsolved problems of Erd\H{o}s in a comprehensive and well-documented volume, the authors hope to continue the work of an unusual and special man who fundamentally influenced the field of mathematics.

Graph Theory as I Have Known It@@@Erdos on Graphs: His Legacy of Unsolved Problems

Multicolor bipartite Ramsey numbers for paths, cycles, and stripes

Directions for Future Research and Summary of Known Values

Bipartite Ramsey numbers of Kt,s in many colors

Star-critical Ramsey number of large cycle and book of different orders

Let $H\xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\}$, and the degree bipartite Ramsey number $br_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\; \mbox{and} \;\chi(H)=2\}$. In this note, we show that $r_{\Delta}(K_{m,n};s)$ is linear on $n$ with $m$ fixed. We also determine $br_{\Delta}(G;s)$ where $G$ are trees, including stars and paths, and complete bipartite graphs.

Degree bipartite Ramsey numbers

Let $H \xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min \{ \Delta(H): H \xrightarrow{s} G \}$, and the degree bipartite Ramsey number $br_{\Delta}(G;s)$ is defined to be $\min \{ \Delta(H): H \xrightarrow{s} G \textrm{ and } \chi(H) = 2 \}$. In this note, we show that $r_{\Delta}(K_{m,n};s)$ is linear on $n$ with fixed $m$. We also evaluate $br_{\Delta}(G;s)$ for paths and other trees.

/pdf/degree-bipartite-ramsey-numbers-yaro71f235.pdf

Degree Bipartite Ramsey Numbers

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Bipartite Ramsey numbers of Kt,s in many colors

Star-critical Ramsey number of large cycle and book of different orders

Degree bipartite Ramsey numbers

Degree Bipartite Ramsey Numbers