Erdős matching conjecture: the four-uniform case

For $r=4$, the manuscript claims the sharp extremal value $$\max\left\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\right\}$$ for every $s\geq1$ and $n\geq4s+4$. Both terms are attained by the standard clique and cover constructions. The claimed advance is an all-parameter four-uniform theorem, extending the sufficiently-large matching-number range of Hou–Hu–Liu ($s\geq6961$) while retaining and crediting their framework and eight computations. The proof combines finite exact certificates and exhaustive searches with a written combinatorial reduction and analytic propagation. Additional equality and stability conclusions have restricted domains; no classification of every extremizer or arbitrary-family stability is claimed. This is a claimed complete r=4 result and a partial result toward the general Erdős matching conjecture. Uniformity five and arbitrary uniformity are not settled by this paper.

For $r=4$, the manuscript claims the sharp extremal value $$\max\left\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\right\}$$ for every $s\geq1$ and $n\geq4s+4$. Both terms are attained by the standard clique and cover constructions. The claimed advance is an all-parameter four-uniform theorem, extending the sufficiently-large matching-number range of Hou–Hu–Liu ($s\geq6961$) while retaining and crediting their framework and eight computations. The proof combines finite exact certificates and exhaustive searches with a written combinatorial reduction and analytic propagation. Additional equality and stability conclusions have restricted domains; no classification of every extremizer or arbitrary-family stability is claimed. This is a claimed complete r=4 result and a partial result toward the general Erdős matching conjecture. Uniformity five and arbitrary uniformity are not settled by this paper.