Inducibility of rainbow graphs

TL;DR

This paper determines the inducibility of rainbow k-cliques for large k, showing it equals a specific ratio, and extends the result to certain connected rainbow graphs with high minimum degree.

Contribution

It proves the inducibility formula for rainbow k-cliques with k ≥ 11 and extends the result to connected rainbow graphs with logarithmic minimum degree.

Findings

Inducibility of rainbow k-cliques is k!/(k^k - k) for k ≥ 11.

Balanced recursive blow-up achieves extremal construction.

High minimum degree ensures the inducibility formula applies.

Abstract

Fix $k\ge 11$ and a rainbow $k$-clique $R$. We prove that the inducibility of $R$ is $k!/(k^k-k)$. An extremal construction is a balanced recursive blow-up of $R$. This answers a question posed by Huang, that is a generalization of an old problem of Erd\H os and S\'os. It remains open to determine the minimum $k$ for which our result is true. More generally, we prove that there is an absolute constant $C>0$ such that every $k$-vertex connected rainbow graph with minimum degree at least $C\log k$ has inducibility $k!/(k^k-k)$.