A Near-Optimal Linear Range for the Erdős Matching Conjecture

Abstract: The Erdős Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\ge2$, there is $s_0(k)$ such that, whenever $s\ge s_0(k)$ and $n\ge(k+1)s$, every $\mathcal{F}\subseteq\binom{[n]}k$ with $ν(\mathcal{F})\le s$ satisfies[ |\mathcal{F}|\le\binom nk-\binom{n-s}k, ]with equality only for the family of all $k$-sets meeting a fixed $s$-set. This lowers the best previous general linear coefficient from $(5k-2)/3$ to $k+1$. Since the two conjectured constructions exchange asymptotic dominance at $n=(ρ_k+o(1))s$ for a coefficient $ρ_k\in(k,k+1)$, our range lies less than one unit above the unavoidable barrier. We also prove a stability theorem showing that cover families are the only near-extremal configurations throughout this range. A key ingredient in our proof is a probabilistic rigidity statement which forces near-extremal fractional covers to be almost integral.

• The paper proves that for every fixed k and sufficiently large s, any k-uniform hypergraph with n≥(k+1)s and matching number at most s has at most the cover-family size, with equality only for a cover family.
• The paper combines a sharp equality-rigid tail inequality, weak hypergraph regularity, fractional-cover methods, stability, and exactification to overcome the gap between fractional and integral matchings.
• The result improves the previous general range from n≥((5k−2)/3)s to n≥(k+1)s and shows this is within less than one coefficient unit of the asymptotic barrier ρ_k, while identifying a maximal-tail conjecture as the key route to further progress.

The problem and the main result

The Erdős Matching Conjecture (EMC), posed in 1965, asserts that a $k$-uniform hypergraph on $n\ge k(s+1)$ vertices with matching number at most $s$ has at most $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$ edges. The two candidates embody distinct extremal mechanisms: the clique family on $k(s+1)-1$ vertices and the cover family of all $k$-sets meeting a fixed $s$-set. The conjecture remains open in general for $k\ge5$. Prior to this work, the strongest uniform cover-side theorem was due to Frankl and Kupavskii, valid for $n\ge\frac{5k-2}{3}s$ [Frankl–Kupavskii 2022].

The paper by Cao, Liu, and Zhang proves the following: for every fixed $k\ge2$ there is $n\ge k(s+1)$0 such that if $n\ge k(s+1)$1, $n\ge k(s+1)$2, and $n\ge k(s+1)$3 satisfies $n\ge k(s+1)$4, then $n\ge k(s+1)$5, with equality only for a cover family. This lowers the general linear coefficient from $n\ge k(s+1)$6 to $n\ge k(s+1)$7, an improvement of order $n\ge k(s+1)$8 in the coefficient.

Near-optimality of the range

The range is essentially best possible among linear ranges. Let $n\ge k(s+1)$9 be the unique solution of $s$0. Comparing leading terms shows the clique and cover constructions exchange asymptotic dominance at $s$1. Consequently no theorem asserting cover-extremality can hold asymptotically below the coefficient $s$2, and the gap between $s$3 and the true barrier is less than one unit of the coefficient. For $s$4 the theorem recovers the known $s$5 result; the genuinely new cases begin at $s$6, though the method is uniform in $s$7.

The transition coefficient also has a probabilistic interpretation: it is the point where the Bernoulli law $s$8 and the two-point law supported on $s$9 achieve equal tail probability $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$0 under mean constraint $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$1. This connects the EMC to the i.i.d. maximal-tail conjecture of Łuczak, Mieczkowska, and Šileikis.

Probabilistic rigidity

The analytic core is a sharp tail inequality with full equality characterization: if $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$2 are independent nonnegative variables with $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$3 and $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$4, then

$$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$5

with equality only when each $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$6 takes values $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$7 and $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$8 with probabilities $$\max\{\binom{k(s+1)-1}{k},\ \binom{n}{k}-\binom{n-s}{k}\}$$9 and $k(s+1)-1$0. The numerical bound was already implied by the small-deviation theorem of Fu et al.; the new contribution is the equality classification, which is indispensable for stability. The proof proceeds by induction, combining a chain-domination statement (extracted from Vlassis–Thomas) that reduces arbitrary mean-one laws to two-point systems, with a direct simplex estimate derived from Grünbaum's centroid inequality via Brunn–Minkowski cap concavity. Equality in Grünbaum's inequality forces the relevant cap to point along a single coordinate direction, isolating the Bernoulli law; deterministic coordinates are excluded through a strict monotonicity argument, and a mean-shifting reduction handles unequal means. A compactness lemma then converts convergence of tail probabilities to weak convergence of the underlying laws toward $k(s+1)-1$1 — the mechanism that later forces near-extremal fractional covers to be almost integral.

From matching constraints to approximate fractional covers

A key obstruction is that $k(s+1)-1$2 does not imply $k(s+1)-1$3 for the fractional matching number. Under the fractional constraint alone, LP duality plus the tail inequality immediately yields the desired density bound up to $k(s+1)-1$4. To bridge the integrality gap, the authors apply weak hypergraph regularity and pass to a bounded reduced $k(s+1)-1$5-graph. Any sufficiently large fractional matching in the reduced graph is rounded greedily into an ordinary matching of $k(s+1)-1$6 using regularity of dense cells; duality then produces a weight function $k(s+1)-1$7 of average at most $k(s+1)-1$8 covering all but $k(s+1)-1$9 edges. The rounding margin is arranged uniformly over the compact parameter interval $k$0.

Stability

Combining these ingredients yields a stability theorem: for fixed $k$1 and $k$2, any family with $k$3 and $k$4 is within $k$5 (symmetric difference) of some cover family $k$6. The proof runs a compactness argument on empirical weight measures: the size hypothesis forces the product tail probability to attain its maximum $k$7, the compactness rigidity lemma forces the empirical measure to converge to $k$8, and hence the high-weight set has size $k$9 while intermediate-weight vertices are $s$0 in number — yielding the structural approximation.

Exactification

Stability leaves an $s$1 error, whereas exact extremality is decided at single-edge scale. The exactification step moves vertices of $s$2 with many missing link edges into the outside part; their number is bounded by $s$3 via a counting argument against $s$4. On the induced local family, a sparse-parameter version of the EMC — assembled from Erdős–Gallai ($s$5), Frankl–Kupavskii, and Bollobás–Daykin–Erdős, with equality cases characterized via the BDE strict bound forcing $s$6 — applies with its own equality statement. A greedy extension then shows any unresolved excess would create a matching of size $s$7: at each step at least $s$8 unused vertices remain outside, guaranteeing an available edge for every non-exceptional vertex of $s$9. This forces the family to equal $k\ge5$0 exactly, completing the proof of the main theorem together with the sparse case handled directly.

Limitations and open questions

Two limitations are explicit. First, the compactness and regularity arguments are qualitative, so the threshold $k\ge5$1 is not effective. Second, the method is confined to $k\ge5$2 because the tail rigidity theorem requires $k\ge5$3. The paper isolates precisely what would remove this restriction: the cover-dominant part of the Łuczak–Mieczkowska–Šileikis maximal-tail conjecture, asserting the same inequality for all $k\ge5$4 with uniqueness of the Bernoulli law for $k\ge5$5 (uniqueness necessarily fails at the endpoint, where the $k\ge5$6-law ties). Proving this conjecture would extend the main theorem to $k\ge5$7 for every fixed $k\ge5$8 — the asymptotically optimal linear range — via the same stability-plus-exactification scheme with adjusted parameters.

Conclusion

The paper establishes the Erdős Matching Conjecture for $k\ge5$9 with large $n\ge\frac{5k-2}{3}s$0, uniformly in $n\ge\frac{5k-2}{3}s$1, improving the best general coefficient from $n\ge\frac{5k-2}{3}s$2 to $n\ge\frac{5k-2}{3}s$3 and proving stability throughout the range. Its principal methodological contribution is the equality-rigid tail inequality and associated compactness principle, which forces near-extremal fractional covers to be almost integral and thereby reconciles the integral matching problem with its LP relaxation. The remaining distance to the barrier $n\ge\frac{5k-2}{3}s$4 is reduced to a single open analytic statement about maximal tail probabilities of sums of i.i.d. variables.