A general finite-board criterion reduces the r-uniform Erdős Matching Conjecture to finite optimization on an (r²+r-1)-vertex board; this is verified exactly for r=4 and s≥6961.
Towards the Erd\H{o}s matching conjecture for 4-uniform hypergraphs: stability and applications
3 Pith papers cite this work. Polarity classification is still indexing.
3
Pith papers citing it
abstract
A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author. In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form.
fields
math.CO 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Proves the vector-space Erdős matching conjecture m_q(n,k,s) equals the maximum of two explicit constructions in the cases k=2, n=(s+1)k, and large n, with stability and t-cover-free extensions.
For sufficiently large n, the maximum spectral radii of M_{k+1}-free and non-trivial intersecting 3-graphs on n vertices are determined and the extremal hypergraphs are characterized.
citing papers explorer
-
A finite-board reduction for the Erd\H{o}s Matching Conjecture and the 4-uniform case via exact certificates
A general finite-board criterion reduces the r-uniform Erdős Matching Conjecture to finite optimization on an (r²+r-1)-vertex board; this is verified exactly for r=4 and s≥6961.
-
An Erd\H{o}s Matching Conjecture for Vector Spaces
Proves the vector-space Erdős matching conjecture m_q(n,k,s) equals the maximum of two explicit constructions in the cases k=2, n=(s+1)k, and large n, with stability and t-cover-free extensions.
-
Spectral Radius Conditions for 3-Uniform Intersecting Families
For sufficiently large n, the maximum spectral radii of M_{k+1}-free and non-trivial intersecting 3-graphs on n vertices are determined and the extremal hypergraphs are characterized.