Cao–Lu–Zhang conjecture on degree-vector -norms of t-intersecting families

Prove that for integers kgeq tgeq1, 1geq dgeq k-1, and real pgeq2, every t-intersecting family Asubseteqbinom{[n]}{k} with ngeq(t+1)(k-t+1) satisfies the degree-vector inequality ||v_d(A)||_p^pleq||v_d(S_T)||_p^p, where S_T={Sinsubseteq[n] choose k:Tsubseteq S} is a t-star for some t-subset Tinsubseteq[n].

Background

The paper concludes by recording a conjecture proposed by Cao, Lu, and Zhang concerning an l_p-norm extension of Erdős–Ko–Rado-type results for t-intersecting uniform set families. A family Asubseteqbinom{[n]}{k} is t-intersecting when every two members intersect in at least t elements, and v_d(A) denotes its d-degree vector.

The conjecture asserts that, in the sharp Erdős–Ko–Rado range ngeq(t+1)(k-t+1), the maximum p-th power of the l_p norm of the d-degree vector is attained by a t-star. The paper notes that the conjecture had already been proved in the cases d=k-1 and t=1, leaving the general parameter range unresolved in the cited formulation.

References

Inspired by these works, Cao, Lu, and Zhang proposed the following conjecture. Let $k\geq t\geq 1$, $k-1\geq d\geq 1$, and let $p\geq2$ be real. If $n\geq(t+1)(k-t+1)$ and $A\subseteq\binom{[n]}k$ is $t$-intersecting, then $$|\mathbf{v}_d(A)|_pp \leq |\mathbf{v}_d(\mathcal{S}_T)|_pp, $$ where $\mathcal{S}_T$ is a $t$-star.

— An Erdős--Ko--Rado theorem for cross-intersecting families in the Euclidean inner product  (2608.17219 - Wan et al., 18 Aug 2026) in Concluding Remarks, Conjecture 6.1 (cited from Cao, Lu, and Zhang)