125 research outputs found
Bipartite Ramsey Numbers and Zarankiewicz Numbers
The bipartite Ramsey number b(m, n) is the minimum b such that any 2-coloring of Kb,b results in a monochromatic Km,m subgraph in the first color or a monochromatic Kn,n subgraph in the second color. The Zarankiewicz number z(m, n; s, t) is the maximum size among Ks,t-free subgraphs of Km,n. In this work, we discuss the intimate relationship between the two numbers as well as propose a new method for bounding the Zarankiewicz numbers. We use the better bounds to improve the upper bound on b(2, 5), in addition we improve the lower bound of b(2, 5) by construction. The new bounds are shown to be 17 ≤ b(2, 5) ≤ 18. Additionally, we apply the same methods to the multicolor case b(2, 2, 3) which has previously not been studied and determine bounds to be 16 ≤ b(2, 2, 3) ≤ 23
The bipartite Ramsey numbers
For the given bipartite graphs , the multicolor bipartite
Ramsey number is the smallest positive integer
such that any -edge-coloring of contains a monochromatic subgraph
isomorphic to , colored with the th color for some . We
compute the exact values of the bipartite Ramsey numbers for
Some geometric structures and bounds for Ramsey numbers
AbstractWe investigate several bounds for both K2,m−K1,n Ramsey numbers and K2,m−K1,n bipartite Ramsey numbers, extending some previous results. Constructions based on certain geometric structures (designs, projective planes, unitals) yield classes of near-optimal bounds or even exact values. Moreover, relationships between these numbers are also discussed
3‐Color bipartite Ramsey number of cycles and paths
The k-colour bipartite Ramsey number of a bipartite graph H is the least integer n for which
every k-edge-coloured complete bipartite graph Kn,n contains a monochromatic copy of H. The
study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and,
independently, by Gy´arf´as and Lehel, who determined the 2-colour Ramsey number of paths. In
this paper we determine asymptotically the 3-colour bipartite Ramsey number of paths and (even)
cycles
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