A problem of Erdos and Sos on 3-graphs

TL;DR

This paper proves a threshold condition in large 3-uniform hypergraphs that guarantees the presence of a specific subgraph, addressing a question posed by Erdos and Sos and establishing the optimal constant 1/4.

Contribution

It establishes a sharp threshold for the minimum induced edge density in large 3-graphs that ensures a K4- subgraph, solving a problem posed by Erdos and Sos.

Findings

Threshold condition guarantees K4- in large 3-graphs.

The constant 1/4 is proven to be optimal.

Provides a new extremal result in hypergraph theory.

Abstract

We show that for every positive epsilon there exist positive delta and n_0 such that every 3-uniform hypergraph on n>=n_0 vertices with the property that every k-vertex subset, where k>=delta*n, induces at least (1/4 + epsilon)*{k \choose 3} edges, contains K4- as a subgraph, where K4- is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdos and Sos. The constant 1/4 is the best possible.