Ramsey for complete graphs with a dropped edge or a triangle

Luis Montejano (1) , Jonathan Chappelon (2) , Jorge Luis Ramírez Alfonsín (2)

1 UNAM - Instituto de Matematicas
2 IMAG - Institut Montpelliérain Alexander Grothendieck

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 (note that K [k,1] is just K k). In this paper we study R(K [k 1 ,t 1 ] ,. .. , K [kr,tr]) (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). We first present a general upper bound (containing the well-known Graham-Rödl upper bound for complete graphs in the particular case when t i = 1 for all i). We then focus our attention when r = 2 and dropped cliques of order 2 and 3 (edges and triangles). We give the exact value for R(K [n,2] , K [4,3]) and R(K [n,3] , K [4,3]) for all n ≥ 2.

Mots clés: Ramsey number; Recursive formula

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

Luis Montejano, Jonathan Chappelon, Jorge Luis Ramírez Alfonsín. Ramsey for complete graphs with a dropped edge or a triangle. Electronic Notes in Discrete Mathematics, 2017, LAGOS'17 – IX Latin and American Algorithms, Graphs and Optimization, 62, pp.21-25. ⟨10.1016/j.endm.2017.10.005⟩. ⟨hal-02053333⟩