For integers n≥ k and 1≤ t<k, determine maxrho( F): F⊆binom[n]k, |F cap F'|≥ t for all F,F'∈ F, where rho( F) is the spectral radius of the adjacency tensor of the k-uniform hypergraph with edge set F. Characterize all extremal families attaining this maximum. In particular, determine for which parameters the complete t-star A_0=F∈binom[n]k:[t]⊆ F is spectrally extremal, and whether the competing extremal families are the Frankl families.

Spectral Erdős–Ko–Rado problem for uniform hypergraphs

The problem asks whether the asymptotic spectral Erdős–Ko–Rado theorem extends to all parameter ranges. Existing work proves the star is optimal only when the ground set is sufficiently large, while smaller or exceptional regimes may have different extremal families.

Known results

Keevash, Lenz, and Mubayi: for fixed $k$ , $t$ , and $\alpha>1$ , sufficiently large $n$ forces the complete $t$ -star to uniquely maximize $\lambda_\alpha$ .
The same work gives stability: sufficiently large spectral radius forces a family to be a $t$ -star.
A separate asymptotic result gives the $t=1$ case for intersecting uniform families, with unique star-type extremality for sufficiently large $n$ .

August 2026 partial theorem

On August 25, 2026, a report on A Sharp Spectral Erdős--Ko--Rado Theorem for Uniform Hypergraphs stated that the paper characterizes competition between the first two Frankl families for $2\leq t<k$ , gives a sufficient threshold for unique $t$ -star extremality, and determines all $t=1$ extremal structures for $n\geq2k$ . The result is explicitly restricted to these ranges and does not settle every $n,k,t$ case.

Current status (as of August 2026): A claimed partial advance covers the stated parameter ranges, but the full problem remains open for the omitted $n,k,t$ regimes.

Spectral Erdős–Ko–Rado problem for uniform hypergraphs

Primary source

Additional references

Progress summary

Solutions 0