On some Zarankiewicz numbers and bipartite Ramsey Numbers for Quadrilateral

Janusz Dybizbański; Tomasz Dzido; Stanisław Radziszowski

On some Zarankiewicz numbers and bipartite Ramsey Numbers for Quadrilateral

Abstract

The Zarankiewicz number z ( m, n ; s, t ) is the maximum number of edges in a subgraph of K m,n that does not contain K s,t as a subgraph. The bipartite Ramsey number b ( n 1 , · · · , n k ) is the least positive integer b such that any coloring of the edges of K b,b with k colors will result in a monochromatic copy of K n i ,n i in the i -th color, for some i , 1 ≤ i ≤ k . If n i = m for all i , then we denote this number by b k ( m ). In this paper we obtain the exact values of some Zarankiewicz numbers for quadrilateral ( s = t = 2), and we derive new bounds for diagonal multicolor bipartite Ramsey numbers avoiding quadrilateral. In particular, we prove that b 4 (2) = 19, and establish new general lower and upper bounds on b k (2).

Authors (3)

Janusz Dybizbański dr
- Uniwersytet Gdański Instytut Informatyki
Tomasz Dzido
- Uniwesytet Gdański Instytut Informatyki
Stanisław Radziszowski

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Category:: Articles
Type:: artykuł w czasopiśmie wyróżnionym w JCR
Published in:: ARS COMBINATORIA no. 119, pages 275 - 287,
ISSN: 0381-7032
Language:: English
Publication year:: 2015
Bibliographic description:: Dybizbański J., Dzido T., Radziszowski S.: On some Zarankiewicz numbers and bipartite Ramsey Numbers for Quadrilateral// ARS COMBINATORIA. -Vol. 119, (2015), s.275-287
Verified by:: Gdańsk University of Technology