Maximal-Tail Conjecture up to the Optimal Threshold

Establish that, for every fixed k≥2 and every 0<x≤1/ρ_k, independent identically distributed random variables Z_1,…,Z_k taking values in [0,1] and satisfying Z_i≤x obey P(Z_1+⋯+Z_k≥1)≤1−(1−x)^k, with equality for 0<x<1/ρ_k only when their common distribution is (1−x)δ_0+xδ_1.

Background

The parameter ρ_k∈(k,k+1) is defined by ρ_kk−(ρ_k−1)k=kk and marks the asymptotic transition between the clique and cover constructions in the Erdős Matching Conjecture. The paper proves the required tail inequality and equality characterization only in the smaller range x≤1/(k+1).

The conjectured extension to x≤1/ρ_k is identified as the missing analytic ingredient for improving the paper’s hypergraph result from n≥(k+1)s to the asymptotically optimal range n≥(ρ_k+ε)s for every fixed ε>0. Equality is expected to be unique below the endpoint, while uniqueness necessarily fails at x=1/ρ_k because the competing law supported on {0,1/k} also attains equality.

References

The following is exactly the cover-dominant part of the i.i.d. maximal-tail conjecture of \L{}uczak, Mieczkowska and \v{S}ileikis . We include the strict equality statement because it is the form required by the stability argument.

Let $0<x\le1/\rho_k$, and let $Z_1,\dots,Z_k$ be independent, identically distributed random variables in $[0,1]$ with $ Z_i\le x$. Then

(Z_1+\cdots+Z_k\ge1)\le q_k(x)=1-(1-x)k.

If $0<x<1/\rho_k$, equality holds if and only if their common law is $\mu_x=(1-x)\delta_0+x\delta_1$.

— A Near-Optimal Linear Range for the Erdős Matching Conjecture  (2608.19118 - Cao et al., 19 Aug 2026) in Section 6, “Toward the optimal threshold,” Conjecture 6.1