Upper bounds and values for $r(K_m,K_n-K_{1,s})$ and $r(K_m-e,K_n-K_{1,s})$

Jonathan Chappelon (1) , Luis Pedro Montejano (1) , Jorge Luis Ramírez Alfonsín (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K_n$, $n\ge N$ there is either a red copy of $G$ or a blue copy of $H$. Let $K_n-K_{1,s}$ be the complete graph on $n$ vertices from which the edges of $K_{1,s}$ are dropped. In this note we present exact values for $r(K_m-K_{1,1},K_n-K_{1,s})$ and new upper bounds for $r(K_m,K_n-K_{1,s})$ in numerous cases. We also present some results for the Ramsey number of Wheels versus $K_n-K_{1,s}$.

Mots clés: Ramsey numbers; edge coloring

Domaines: Combinatoire [math.CO]

Jonathan Chappelon, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín. Upper bounds and values for $r(K_m,K_n-K_{1,s})$ and $r(K_m-e,K_n-K_{1,s})$. 2013. ⟨hal-00872909v1⟩