Hypergraph Turán Problems in $\ell_2$-Norm
József Balogh, Felix Christian Clemen, Bernard Lidický
TL;DR
The paper investigates extremality in $3$-uniform hypergraphs through the lens of the codegree squared function $exco_2(n,H)$ and contrasts it with classical Turán-type concepts such as $\pi$, $\pi_2$, and $\pi_u$; it develops an integrated approach combining flag algebras and stability to determine asymptotically $exco_2(n,H)$ for several families, including matchings, stars, loose paths and cycles, and the $5$-vertex hypergraph $F_5$, with exact results for $F_5$ and related structures. It also surveys bounds and conjectures for the various densities, introduces extremal constructions like $S_n$ and the balanced complete bipartite graphs, and explains how the flag-algebra framework yields sharp or near-sharp bounds, often with computer-assisted proofs. The article situates these results within a broader program to understand how $\ell_2$-norm extremality interacts with other Turán-type notions, and it discusses open problems and potential extensions to higher uniformities and induced variants. It is presented as a survey with conjectures and plans for regular updates on arXiv, providing a roadmap for extending $\sigma$-type analyses to more complex hypergraph families.
Abstract
There are various different notions measuring extremality of hypergraphs. In this survey we compare the recently introduced notion of the codegree squared extremal function with the Turán function, the minimum codegree threshold and the uniform Turán density. The codegree squared sum $\textrm{co}_2(G)$ of a $3$-uniform hypergraph $G$ is defined to be the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $x,y$. In other words, this is the square of the $\ell_2$-norm of the codegree vector. We are interested in how large $\textrm{co}_2(G)$ can be if we require $G$ to be $H$-free for some $3$-uniform hypergraph $H$. This maximum value of $\textrm{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$ is called the codegree squared extremal function, which we denote by $\textrm{exco}_2(n,H)$. We systemically study the extremal codegree squared sum of various $3$-uniform hypergraphs using various proof techniques. Some of our proofs rely on the flag algebra method while others use more classical tools such as the stability method. In particular, we (asymptotically) determine the codegree squared extremal numbers of matchings, stars, paths, cycles, and $F_5$, the $5$-vertex hypergraph with edge set $\{123,124,345\}$. Additionally, our paper has a survey format, as we state several conjectures and give an overview of Turán densities, minimum codegree thresholds and codegree squared extremal numbers of popular hypergraphs. We intend to update the arXiv version of this paper regularly.