Given integers $r > t \ge 1$ and a real number $p > 0$, the $(t,p)$-norm $\left\lVert \mathcal{ H} \right\rVert_{ t,p}$ of an $r$-graph $\mathcal{ H}$ is the sum of the $p$-th power of the degrees $d_{ \mathcal{ H}}(T)$ over all $t$-subsets $T \subset V(\mathcal{ H})$. We conduct a systematic study of the Tur\'{ a}n-type problem of determining $\mathrm{ ex}_{ t,p}(n,\mathcal{ F})$, which is the maximum of $\left\lVert \mathcal{ H} \right\rVert_{ t,p}$ over all $n$-vertex $\mathcal{ F}$-free $r$-graphs $\mathcal{ H}$. We establish several basic properties for the $(t,p)$-norm of $r$-graphs, enabling us to derive general theorems from the recently established framework in~\cite{ CL24} that are useful for determining $\mathrm{ ex}_{ t,p}(n,\mathcal{ F})$ and proving the corresponding stability. We determine the asymptotic value of $\mathrm{ ex}_{ t,p}(n,H_{ F}^{ r})$ for all feasible combinations of $(r,t,p)$ and for every graph $F$ with chromatic number greater than $r$, where $H_{ F}^{ r}$ represents the expansion of $F$. In the case where $F$ is edge-critical and $p \ge 1$, we establish strong stability and determine the exact value of $\mathrm{ ex}_{ t,p}(n,H_{ F}^{ r})$ for all sufficiently large $n$. These results extend the seminal theorems of Erd\H{ o}s--Stone--Simonovits, Andr\'{ a}sfai--Erd\H{ o}s--S\'{ o}s, Erd\H{ o}s--Simonovits, and a classical theorem of Mubayi. For the $3$-uniform generalized triangle $F_5$, we determine the exact value of $\mathrm{ ex}_{ 2,p}(n,F_5)$ for all $p \ge 1$ and its asymptotic value for all $p \in [1/2, 1]\cup \{ k^{ -1} \colon k \in 6\mathbb{ N}^{ +}+\{ 0,2\}\}$. This extends old theorems of Bollob\'{ a}s, Frankl--F\"{ u}redi, and a recent result of Balogh--Clemen--Lidick\'{ y}. Our proofs utilize results on the graph inducibility problem, Steiner triple systems, and the feasible region problem introduced by Liu--Mubayi.
Categories: math.CO