NDLI: Asymptotic Size Ramsey Results for Bipartite Graphs

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Asymptotic Size Ramsey Results for Bipartite Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Pikhurko, Oleg
Copyright Year	2003
Abstract	We show that $\lim_{n\to\infty}\hat r(F_{1,n},\dots,F_{q,n},F_{q+1},\dots,F_{r})/n$ exists, where the bipartite graphs $F_{q+1},\dots,F_r$ do not depend on n while, for $1\le i\le q$, $F_{i,n}$ is obtained from some bipartite graph $F_i$ with parts $V_1\cup V_2=V(F_i)$ by duplicating each vertex $v\in V_2$ $(c_v+o(1))n$ times for some real $c_v > 0$.In fact, the limit is the minimum of a certain mixed integer program. Using the Farkas lemma we show how to compute it when each forbidden graph is a complete bipartite graph, in particular answering the question of Erdos, Faudree, Rousseau, and Schelp [Period.\ Math.\ Hungar., 9 (1978), pp. 145--161], who asked for the asymptotics of $\hat r(K_{s,n},K_{s,n})$ for fixed s and large n. Also, we prove (for all sufficiently large n) the conjecture of Faudree, Rousseau, and Sheehan in [Graph Theory and Combinatorics, B. Bollobas, ed., Cambridge University Press, Cambridge, UK, 1984, pp. 273--281] that $\hat r(K_{2,n},K_{2,n}) =18n-15$.
Starting Page	99
Ending Page	113
Page Count	15
File Format	PDF
ISSN	08954801
DOI	10.1137/S0895480101384086
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	16
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-01
Access Restriction	Subscribed
Subject Keyword	bipartite graphs Generalized Ramsey theory size Ramsey number mixed integer programming Mixed integer programming Linear programming Farkas lemma Extremal problems
Content Type	Text
Resource Type	Article
Subject	Mathematics

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