# Approximate Steiner $(r-1,r,n)$-systems without $3$ blocks on $r+2$ points

## TL;DR

This paper investigates the maximum size of certain hypergraphs avoiding specific small configurations, providing asymptotic bounds for these extremal problems in combinatorics.

## Contribution

It establishes asymptotic bounds for the maximum number of edges in hypergraphs avoiding particular small subgraphs, advancing understanding of approximate Steiner systems.

## Key findings

- Asymptotic maximum edges are approximately (1/r) * binomial(n, r-1).
- Provides bounds for hypergraphs avoiding small configurations.
- Advances extremal combinatorics in hypergraph theory.

## Abstract

For a family ${\mathcal F}$ of $r$-graphs, let $\mathrm{ex}(n,{\mathcal F})$ denote the maximum number of edges in an ${\mathcal F}$-free $r$-graph on $n$ vertices. Let ${\mathcal F}_r(v,e)$ denote the family of all $r$-graphs with $e$ edges and at most $v$ vertices. We prove that $\mathrm{ex}(n,{\mathcal F}_r(r+1,2) \cup {\mathcal F}_r(r+2,3)) = (\frac{1}{r} - o(1)) \binom{n}{r-1}$.