기초과학연구원 이산수학그룹
On November 11, 2025, Simón Piga from the University of Hamburg gave a talk at the Discrete Math Seminar on variants of the Turán problem for k-uniform hypergraphs. The title of his talk was “Turán problem in hypergraphs with quasirandom links“.
Given a $k$-uniform hypergraph $F$, its Turán density $\pi(F)$ is the infimum over all $d\in [0,1]$ such that any $n$-vertex $k$-uniform hypergraph $H$ with at least $d\binom{n}{k}+o(n^k)$ edges contains a copy of $F$. While Turán densities are generally well understood for graphs ($k=2$), the problem becomes notoriously difficult for $k\geq 3$, even for small hypergraphs.
We study two well-known variants of this Turán problem for hypergraphs: first, under minimum codegree conditions and, second, with a quasirandom edge distribution. Each variant defines a distinct extremal parameter, generalising the classical Turán density. Here we present recent results in both settings, with a particular emphasis on the case of hypergraphs where every link is itself quasirandom. Our results include exact solutions for key hypergraphs and general results about the behaviour of the Turán density functions.