The $(t,p)$-Norm in Classical Extremal Problems

Abstract

Given integers $r>t\ge1$ and a real number $p>0$, the $(t,p)$-norm $||\mathcal{H}||_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th powers of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T\subseteq V(\mathcal{H})$. 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 Erd\H{o}s--Ko--Rado-type theorem for the $(t,p)$-norm. Third, for $P_\ell^r$-free hypergraphs, we determine the maximum $(t,p)$-norm for every $1\le t\le r-1$ and $p>1$. In each of the three settings, we also characterize all extremal families.