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Some Ramsey Schur Numbers

Published online by Cambridge University Press:  15 February 2005

JENS-P. BODE ,
HANS-DIETRICH O. F. GRONAU  and
HEIKO HARBORTH
JENS-P. BODE
Affiliation:
Diskrete Mathematik, Technische Universität Braunschweig, 38023 Braunschweig, Germany (e-mail: jp.bode@tu-bs.de, h.harborth@tu-bs.de)
HANS-DIETRICH O. F. GRONAU
Affiliation:
Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, 18055 Rostock, Germany (e-mail: gronau@mathematik.uni-rostock.de)
HEIKO HARBORTH
Affiliation:
Diskrete Mathematik, Technische Universität Braunschweig, 38023 Braunschweig, Germany (e-mail: jp.bode@tu-bs.de, h.harborth@tu-bs.de)
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Abstract

The Ramsey Schur number $RS(s,t)$ is the smallest $n$ such that every 2-colouring of the edges of $K_n$ with vertices $1,2,\ldots,n$ contains a green $K_s$ or there are vertices $x_1,x_2,\ldots,x_t$ fulfilling the equation $x_1+x_2+\cdots+x_{t-1}=x_t$ and all edges $(x_i,x_j)$ are red. We prove $RS(3,3)=11, RS(3,t)=t^2-3$ for $t\equiv1\ (\mbox{mod}\ 6)$ and $t=8$, and $RS(3,t)\geq t^2-3$.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 14 , Issue 1-2 , January 2005 , pp. 25 - 30
Copyright
© 2005 Cambridge University Press

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