The inducibility of Turán graphs

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 Turán graphs $F$. This topic has been extensively studied in the literature, including works of Pippenger--Golumbic, Brown--Sidorenko, Bollobás--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 Turán graph $F$ and sufficiently large $n$, the value $I(F,n)$ is attained uniquely by the $m$-partite Turán 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 Bollobás--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.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.3
• Definition 1.4: LiuPikhurko23
• Theorem 1.5
• Lemma 2.2: Brown94
• Lemma 2.3
• Definition 2.4: LiuPikhurko23
• Theorem 2.5: LiuPikhurko23
• Theorem 3.1
• Lemma 3.2