2024/06/22 by Wanfang Chen, Daniel Iľkovič, Chen, Wanfang +7 · 2 citations
Mathematics · Engineering · #Differential Equations and Boundary Problems #Material Science and Thermodynamics
paper · pdf · doi:10.48550/arxiv.2406.15934
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 power of the degrees dH(T) over all t-subsets T ⊂ V(H). We conduct a systematic study of the Turán-type problem of determining ext,p(n,F), which is the maximum of ‖ H ‖t,p over all n-vertex F-free r-graphs 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~\citeCL24 that are useful for determining ext,p(n,F) and proving the corresponding stability. We determine the asymptotic value of ext,p(n,HFr) for all feasible combinations of (r,t,p) and for every graph F with chromatic number greater than r, where HFr represents the expansion of F. In the case where F is edge-critical and p ≥ 1, we establish strong stability and determine the exact value of ext,p(n,HFr) for all sufficiently large n. These results extend the seminal theorems of Erdős--Stone--Simonovits, Andrásfai--Erdős--Sós, Erdős--Simonovits, and a classical theorem of Mubayi. For the 3-uniform generalized triangle F5, we determine the exact value of ex2,p(n,F5) for all p ≥ 1 and its asymptotic value for all p ∈ [1/2, 1]∪ \k-1 \colon k ∈ 6ℕ++\0,2\\. This extends old theorems of Bollobás, Frankl--Füredi, and a recent result of Balogh--Clemen--Lidický. Our proofs utilize results on the graph inducibility problem, Steiner triple systems, and the feasible region problem introduced by Liu--Mubayi.