Generalized Tur\'an problem for Complete Hypergraphs

TL;DR

This paper establishes an asymptotically tight upper bound for the maximum density of smaller complete hypergraphs within larger hypergraphs that avoid even larger complete hypergraphs, extending Turán-type results to hypergraphs.

Contribution

It provides the first asymptotic upper bound for the generalized Turán problem in hypergraphs using flag algebra techniques, matching known bounds in special cases.

Findings

Derived an asymptotic upper bound for hypergraph Turán densities.

Applied flag algebra methods to establish linear relations between densities.

Provided a flag algebra certificate for a specific hypergraph density limit.

Abstract

Write $K^{(k)}_{n}$ for the complete $k$-graph on $n$ vertices. For $2 \leq k \leq g < r$ integers, let $\pi\left(n, K^{(k)}_{g}, K^{(k)}_r\right)$ be the maximum density of $K^{(k)}_{g}$ in $n$ vertex $K^{(k)}_{r}$-free $k$-graphs. The main contribution of this paper is the upper bound: $\pi\left(n, K^{(k)}_{g}, K^{(k)}_r\right) \leq \left(1 + O\left(n^{-1}\right) \right)\prod_{m=k}^{g} \left(1 - \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}} \right).$ The graph case ($k=2$) is the first known generalized Tur\'an question, investigated by Erd\H{o}s. The $k=g$ case is the hypergraph Tur\'an problem where the best known general upper bound is by de Caen. The result proved here matches both bounds asymptotically, while any triple $k, g, r$ with $2 < k < g < r$ provides a new upper bound. The proof uses techniques from the theory of flag algebras to derive linear relations between different densities. These relations can be combined with linear algebraic methods. Additionally a simple flag algebraic certificate will be given for $\lim_{n \rightarrow \infty} \pi \left(n, K^{(3)}_4, K^{(3)}_5 \right) = 3/8$.