Geodesic
Subgraph densities in hypergraphs
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 2, pp. 281-297.

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Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if for any ε > 0 and any integer m ≥ r, any r-uniform graph with n > n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on ε and m. A result of Erdös, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. Erdös asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumps for every r ≥ 3. However, there are still a lot of open questions on determining whether or not a number is a jump for r ≥ 3. In this paper, we first find an infinite sequence of non-jumps for r = 4, then extend one of them to every r ≥ 4. Our approach is based on the techniques developed by Frankl and Rödl.
Keywords: Erdös jumping constant conjecture, Lagrangian, optimal vector 
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Peng, Yuejian. Subgraph densities in hypergraphs. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 2, pp. 281-297. http://geodesic.mathdoc.fr/item/DMGT_2007_27_2_a5/

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