The semi-inducibility of the blue--blue--red path on four vertices

Abstract

For an $n$-vertex graph $G$, let $N(H_3,G)$ be the number of injective labeled copies of the red-blue path $H_3$ for which the two blue pairs are mapped to non-edges of $G$ and the red pair is mapped to an edge of $G$. We determine the maximum limiting value of $N(H_3,G)/n^4$ and give an extremal construction, which is the disjoint union of a clique and an asymptotically regular graph. The proof uses weighted vertex quotients and degree-square tie-breaking. We thereby resolve the exceptional four-vertex case left open in the recent classification of non-complete red-blue graphs.

Disclosure

“

= F (a∗ , x∗ ) + o(1) = p∗ + o(1). Finally, qn /mn = x∗ /(1 − a∗ ) + o(1) = λ∗ + o(1), which gives the asserted relative degree inside Rn . 6 Declaration on the use of AI The author used generative AI tools to assist with discussing proof strategies, proof checking, and exposition. All mathematical arguments, results, and conclusions were reviewed and verified by the author. References [BLM+26] J. Balogh, B. Lidický, D. Mubayi, F. Pfen”

PDF page 22

Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:15:11.158452Z

Structural counts