Some Ramsey Schur Numbers

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(...

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Veröffentlicht in:Combinatorics, probability & computing probability & computing, 2005-01, Vol.14 (1-2), p.25-30
Hauptverfasser: BODE, JENS-P., GRONAU, HANS-DIETRICH O. F., HARBORTH, HEIKO
Format: Artikel
Sprache:eng
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Zusammenfassung: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 $RS(3,t)\geq t^2-3$.
ISSN:0963-5483
1469-2163
DOI:10.1017/S0963548304006595