Quantitative upper bound for symmetric 3-wise intersecting families

Prove that there exist universal constants c,δ>0 such that every symmetric 3-wise intersecting family A⊆𝒫_n satisfies log₂|A|≤n−cn^δ, while recognizing that no such result can hold with δ>1/2.

Background

The paper discusses the gap between the known asymptotic result that symmetric 3-wise intersecting families have size o(2n) and the substantially stronger quantitative behavior suggested by constructions. A construction of Riordan achieves log₂|A|=n−2√n+o(√n) for infinitely many n, indicating that an exponent δ greater than 1/2 cannot be valid in a universal bound of the stated form.

The conjecture is attributed to Ellis and Narayanan. The paper’s results establish quantitative bounds for regular increasing 3-wise intersecting families, but these bounds occur at the weaker scale |A|/2n≲√(log n/n), so they do not resolve the conjectured stretched-exponential deficit for symmetric families.

References

A construction of Riordan, recorded in, gives symmetric $3$-wise intersecting families satisfying $$ \log_2|A|=n-2\sqrt n+o(\sqrt n) $$ for infinitely many $n$, and therefore they conjectured that every symmetric 3-wise intersecting family $A\subseteq\mathcal P_n$ satisfies $$ \log_2|A|\le n-cn\delta $$ for some universal constants $c,\delta>0$ and one cannot take $\delta >1/2$ in such a result.

— Quantitative bounds for regular $3$-wise intersecting families  (2608.20242 - Chang, 20 Aug 2026) in Section 1, Introduction; reiterated in Section 5, Concluding remarks