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Claim. The preprint The Erdős Matching Conjecture for 4-uniform hypergraphs by Jianfeng Hou, Caiyun Hu and Xizhi Liu, arXiv:2605.26060 (posted 2026-05-25, revised 2026-09-11), states that for integers $s\ge6004$ and $n\ge4s+4$, an $n$-vertex $4$-uniform hypergraph with no matching of size $s+1$ has at most $\max\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\}$ edges. In the notation of Problem 1020, with $k=s+1$,

$$f(n;4,k)=\max\left(\binom{4k-1}{4},\binom n4-\binom{n-k+1}{4}\right) \qquad(k\ge6005,\ n\ge4k).$$

The abstract describes a finite-board reduction for general uniformity, which reduces the conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the two candidate constructions exchange dominance; in the $4$-uniform case the board has $19$ vertices, its weighted inequality splits into layers of $19$, $15$ and $11$ vertices, the hardest layer is settled by exact rational dual certificates and deterministic integer searches, and every computer-assisted step is checked in exact arithmetic by verifiers that rebuild the finite systems from their definitions.

Covers. The case $r=4$ for $k\ge6005$ and $n\ge4k$. The matching numbers below $6004$ are the claimed novelty of Babanskyy 2026, which develops this preprint's method; the range $n\ge5(k-1)$ with $n$ large is the earlier claim on Frankl, Lu, Ma and Wu 2026.

Depends on. No page of this wiki.

Standing. Claimed. The preprint is not refereed, it was not posted on the site's discussion thread or proof-claims tab, and the site's label and commentary, last edited on 28 December 2025, do not mention it. The proof, including its computer-assisted steps, is not verified by this corpus.