On the limit of the positive ℓ-degree Turán problem

2023/02/16 by Oleg Pikhurko, Pikhurko, Oleg
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2302.08123

openalex publication_date 2023/02/16 · openalex created_date 2023/02/18 · openalex updated_date 2026/07/28

Abstract

The minimum positive ℓ-degree δ⁺_ℓ(G) of a non-empty k-graph G is the maximum m such that every ℓ-subset of V(G) is contained in either none or at least m edges of G; let δ⁺_ℓ(G):=0 if G has no edges. For a family \mathcal F of k-graphs, let co⁺ex_ℓ(n,\mathcal F) be the maximum of δ⁺_ℓ(G) over all \mathcal F-free k-graphs G on n vertices. We prove that the ratio co⁺ex_ℓ(n,\mathcal F)/n-ℓ\choose k-ℓ tends to limit as n→∞, answering a question of Halfpap, Lemons and Palmer. Also, we show that the limit can be obtained as the value of a natural optimisation problem for k-hypergraphons; in fact, we give an alternative description of the set of possible accumulation points of almost extremal k-graphs.

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