Abstract:We prove the 4-uniform Erdős Matching Conjecture for every matching number $s\ge 6961$. The proof has two parts. First, building on ideas from Frankl--Rödl--Ruciński, we formulate a general finite-board criterion for the $r$-uniform conjecture. The criterion has two assumptions: the $(r-1)$-uniform cover-side bound for links with matching number at most $t$ holds at every $m\ge n_r(t)$, and a finite optimization problem for mixed-size trace configurations on an $(r^2+r-1)$-vertex board. Together with the corresponding lower-uniformity input, this finite-board optimization implies the Erdős Matching Conjecture with explicit large-matching thresholds.
Second, we verify the finite-board assumption for $r=4$. The local board has 19 vertices, and the required inequality is decomposed into three weighted local inequalities: a leading wide layer, a 15-board layer, and an 11-board layer. The verification is reduced to exact finite optimization and certificate-validation problems: Ferrers down-set enumerations for pair and triple traces, rational Farkas-dual certificates for the top-star branch, integer branch-and-bound up-set hitting and pattern searches for the no-top-star branch, and residual-cut dual certificates for the 15-board and 11-board layers.
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.26060 [math.CO] |
| (or arXiv:2605.26060v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.26060 arXiv-issued DOI via DataCite (pending registration) |
From: Xizhi Liu [view email]
[v1]
Mon, 25 May 2026 17:19:09 UTC (32 KB)
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