Inducibility of rainbow graphs

TL;DR

The paper determines the inducibility of rainbow $k$-cliques, proving that for $k\ge 11$ the rainbow $K_k$ is a fractalizer with inducibility $\frac{k!}{k^k-k}$, realized by the iterated balanced blow-up construction. It develops a general framework showing that any connected rainbow graph on $k$ vertices with minimum degree at least a constant times $\log k$ also attains this inducibility, thus extending the fractalizer phenomenon beyond tournaments. The results connect to the historical Erdős–Sós problem and the PG conjecture in colored and undirected settings, providing both a concrete threshold and a robust technique (partition- and distance-based analyses with Zykov-type regularization) for inducibility in rainbow structures. The work also clarifies that disconnected rainbow graphs are not fractalizers and delineates the boundary between dense and sparse regimes in this theory, with implications for broader extremal questions in colored graphs.

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 Sós. 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 $k!/(k^k-k)$.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1.2: MR
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4