Obtain reasonably small threshold bounds for the Hajnal–Rothschild theorem

Establish reasonably small upper bounds for the threshold \(n_0(k,t,s)\) in the Hajnal–Rothschild theorem, valid for general values of \(k\), \(t\), and \(s\).

Background

The Hajnal–Rothschild theorem asserts that sufficiently large n forces every k-uniform family with $\nu(\mathcal F,t)\le s$ to have size at most the matching-based construction $h'(n,k,t,s)$. The threshold $n_0(k,t,s)$ specifies how large n must be for this conclusion to hold.

The paper contrasts the known relatively small bounds in the Erdős Matching Conjecture case $t=1$ with the much larger bounds arising from the general Hajnal–Rothschild proof. Improving these bounds would make the theorem effective over a substantially broader parameter range.