Minimum degree conditions for rainbow triangles

Klas Markström, Victor Falgas-Ravry, Eero Räty

Publication date: 22 May 2023

Abstract: Let

m a t h b f G : = (G_{1}, G_{2}, G_{3})

be a triple of graphs on a common vertex set

V

of size

n

. A rainbow triangle in

m a t h b f G

is a triple of edges

(e_{1}, e_{2}, e_{3})

with

e_{i} i n G_{i}

for each

i

and

e_{1}, e_{2}, e_{3}

forming a triangle in

V

. In this paper we consider the following question: what triples of minimum degree conditions

(d e l t a (G_{1}), d e l t a (G_{2}), d e l t a (G_{3}))

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, Montejanos and v{S}'amal 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.

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