Local maximum of inducibility profiles

TL;DR

This paper investigates the inducibility profile of a specific graph, demonstrating the existence of multiple local maxima and explicitly determining the profile at certain edge densities.

Contribution

It proves that the inducibility profile of K_{2,2,1} has multiple local maxima and provides explicit calculations at specific edge densities.

Findings

I_{K_{2,2,1}}(e) has at least two local maxima in (0,1).

Explicit values of I_{K_{2,2,1}}(e) are determined for e=(k-1)/k, k≥3.

Abstract

For a graph $G$ and $e\in [0,1]$, denote by $I_G(e)$ the supremum of densities of $G$ over $n$-vertex graphs with edge density $e$ as $n$ goes to infinity. Liu, Mubayi and Reiher asked if there exists a graph $G$, where $I_G(e)$ has a non-trivial local maximum. In this note we resolve their problem by showing that $I_{K_{2,2,1}}(e)$ has at least two local maxima in $(0,1)$. Additionally, we determine $I_{K_{2,2,1}}(e)$, when $e=(k-1)/k$ for every integer $k\ge 3.$