2026/08/05 by Xiamiao Zhao, Yuanpei Wang
Mathematics · #math.CO
arxiv created 2026/08/05 · arxiv updated 2026/08/06
Given integers r>t≥1 and a real number p>0, the (t,p)-norm ||H||t,p of an r-graph H is the sum of the p-th powers of the degrees dH(T) over all t-subsets T⊆ V(H). When t=r-1, this is the codegree p-norm. For all sufficiently large n, we obtain the following results. The first two apply in both the convex range p>1 and the concave range 0<p<1. First, for r-graphs with matching number at most s, we determine the maximum (t,p)-norm. Second, for k-intersecting families, we establish an Erdős--Ko--Rado-type theorem for the (t,p)-norm. Third, for P_ℓr-free hypergraphs, we determine the maximum (t,p)-norm for every 1≤ t≤ r-1 and p>1. In each of the three settings, we also characterize all extremal families.