Disconnected Common Graphs via Supersaturation

Jae-baek LeeJonathan A. Noel

Mar 2023

摘要

A graph $H$ is said to be "common" if the number of monochromatic labelledcopies of $H$ in a $2$-colouring of the edges of a large complete graph isasymptotically minimized by a random colouring. It is well known that thedisjoint union of two common graphs may be uncommon; e.g., $K_2$ and $K_3$ arecommon, but their disjoint union is not. We find the first examples of a pairof uncommon graphs whose disjoint union is common and a common graph and anuncommon graph whose disjoint union is common. Our approach is to reduce theproblem of showing that certain disconnected graphs are common to a constrainedoptimization problem in which the constraints are derived from supersaturationbounds related to Razborov's Triangle Density Theorem. We also improve boundson the Ramsey multiplicity constant of a triangle with a pendant edge and thedisjoint union of $K_3$ and $K_2$.

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