Minimum‐degree conditions for rainbow triangles

Abstract

AbstractLet be a triple of graphs on a common vertex set of size . A rainbow triangle in is a triple of edges with for each and forming a triangle in . In this paper we consider the following question: what triples of minimum‐degree conditions guarantee the existence of a rainbow triangle? This may be seen as a minimum‐degree version of a problem of Aharoni, DeVos, de la Maza, Montejano and Šámal on density conditions for rainbow triangles, which was recently resolved by the authors. We establish that the extremal behaviour in the minimum‐degree setting differs strikingly from that seen in the density setting, with discrete jumps as opposed to continuous transitions. Our work leaves a number of natural questions open, which we discuss.