Morrison–Nir–Norin–Rzążewski–Wesolek Turán-goodness questions

For a graph $H$ and integer $r\ge 2$ , say that $H$ is $K_{r+1}$ -Turán-good if, for every sufficiently large $n$ , the Turán graph $T_r(n)$ maximizes the number of copies of $H$ among all $n$ -vertex $K_{r+1}$ -free graphs. The problems are: (1) determine whether there is an absolute constant $C$ such that every graph $H$ is $K_{r+1}$ -Turán-good whenever $r\ge C v(H)^2$ ; and (2) determine whether Turán-goodness is monotone in the forbidden clique size, namely whether, for every graph $H$ and every integer $r\ge 2$ , $H$ being $K_r$ -Turán-good implies that $H$ is $K_{r+1}$ -Turán-good.

Progress summary

Solved

A new unrefereed preprint claims to answer both questions, but its proof has not yet been independently verified.

The questions ask how far the threshold for Turán-goodness can be improved and whether Turán-goodness is monotone in the forbidden clique size. The earlier eventual-goodness conjecture of Gerbner and Palmer was proved by Morrison, Nir, Norin, Rzążewski, and Wesolek in 2022.

Known results

Every graph $H$ is eventually $K_{r+1}$ -Turán-good (Morrison, Nir, Norin, Rzążewski, and Wesolek, 2022).
Every graph $H$ is $K_{r+1}$ -Turán-stable for $r\ge 300v(H)^9$ (2023 account of the earlier result).

August 2026 threshold and monotonicity claims

Wang, Kang, and Zhao claim that every graph with an edge is strictly $K_{r+1}$ -Turán-good and $K_{r+1}$ -Turán-stable for $r\ge 168e(H)$ , improving the earlier threshold. They also claim that for every $r\ge 3$ some graph is $K_r$ -Turán-good but not $K_{r+1}$ -Turán-good, with $h-2\sqrt h-O(1)\le\lambda_{\max}(h)\le84h^2$ ; whether the threshold is $(1+o(1))h$ remains open.

Current status (as of August 2026): The 2026 preprint claims the threshold and monotonicity questions are settled up to the stated bounds, but the proofs remain unverified and the sharp monotonicity threshold is open.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Eventually Turán good I: Edge-Linear Thresholds and Monotonicity — arXiv — Yuanpei Wang, Liying Kang, Xiamiao Zhao

extremal-graph-theory

1 day ago