Ramsey for complete graphs with dropped cliques

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

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Let $K_{[k,t]}$ be the complete graph on $k$ vertices from which a set of edges, induced by a clique of order $t$, has been dropped. In this note we give two explicit upper bounds for $R(K_{[k_1,t_1]},\dots, K_{[k_r,t_r]})$ (the smallest integer $n$ such that for any $r$-edge coloring of $K_n$ there always occurs a monochromatic $K_{[k_i,t_i]}$ for some $i$). Our first upper bound contains a classical one in the case when $k_1=\cdots =k_r$ and $t_i=1$ for all $i$. The second one is obtained by introducing a new edge coloring called $\chi_r$-colorings. We finally discuss a conjecture claiming, in particular, that our second upper bound improves the classical one in infinitely many cases.

Mots clés: recursive formula; Ramsey number

Domaines: Combinatoire [math.CO]; Mathématiques [math]

Jonathan Chappelon, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín. Ramsey for complete graphs with dropped cliques. 2014. ⟨hal-00847564v3⟩