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arXiv · 2601.10548

The inducibility of Tur\'an graphs

Xizhi Liu·Jie Ma·Tianming Zhu
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Abstract

Let $I(F,n)$ denote the maximum number of induced copies of a graph $F$ in an $n$-vertex graph. The inducibility of $F$, defined as $i(F)=\lim_{n\to \infty} I(F,n)/\binom{n}{v(F)}$, is a central problem in extremal graph theory. In this work, we investigate the inducibility of Tur\'an graphs $F$. This topic has been extensively studied in the literature, including works of Pippenger--Golumbic, Brown--Sidorenko, Bollob\'as--Egawa--Harris--Jin, Mubayi, Reiher, and the first author, and Yuster. Broadly speaking, these results resolve or asymptotically resolve the problem when the part sizes of $F$ are either sufficiently large or sufficiently small (at most four). We complete this picture by proving that for every Tur\'an graph $F$ and sufficiently large $n$, the value $I(F,n)$ is attained uniquely by the $m$-partite Tur\'an graph on $n$ vertices, where $m$ is given explicitly in terms of the number of parts and vertices of $F$. This confirms a conjecture of Bollob\'as--Egawa--Harris--Jin from 1995, and we also establish the corresponding stability theorem. Moreover, we prove an asymptotic analogue for $I_{k+1}(F,n)$, the maximum number of induced copies of $F$ in an $n$-vertex $K_{k+1}$-free graph, thereby completely resolving a recent problem of Yuster. Finally, our results extend to a broader class of complete multipartite graphs in which the largest and smallest part sizes differ by at most on the order of the square root of the smallest part size.

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Xizhi Liu, Jie Ma, Tianming Zhu. 2026-01-15. The inducibility of Tur\'an graphs. https://arxiv.org/abs/2601.10548

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