The size of a hypergraph and its matching number

TL;DR

This paper verifies Erdős's conjecture on the maximum size of K-uniform hypergraphs without T disjoint edges for a broader range of T, advancing understanding of hypergraph Turán problems.

Contribution

The paper proves Erdős's conjecture for all T < N/(3K^2), significantly extending the previously known range T = O(N/K^3).

Findings

Confirmed Erdős's conjecture for T < N/(3K^2)

Improved the known bounds on hypergraph sizes without T disjoint edges

Extended the range of T for which the conjecture holds

Abstract

More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a basic instance of the hypergraph Tur\'an problem (with a T-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all T < N/(3K^2). This improves upon the best previously known range T = O(N/K^3), which dates back to the 1970's.