Let r ≥ 2 be an integer.The real number α E[0,1)is a jump for r if there exists a constant c＞0 such that for any ε＞0 and any integer m ≥ r,there exists an integer n0(ε,m)satisfying 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 leastα+c.A result of Erdös and Simonovits(1966)and Erdös and Stone(1946)implies that every α ∈[0,1)is a jump for r=2.Erdös(1964)asked whether the same is true for r≥3.Frankl and Rödl(1984)gave a negative answer by showing that 1-1/ℓr-1is not a jump for r if r≥3 and ℓ＞2r.After that,more non-jumps are found by using a method of Frankl and Rödl(1984).Motivated by an idea of Liu and Pikhurko(2023),in this paper,we show a method to construct maps f:[0,1)→[0,1)that preserve non-jumps,i.e.,if α is a non-jump for r given by the method of Frankl and Rödl(1984),then f(α)is also a non-jump for r.We use these maps to study hypergraph Turán densities and answer a question posed by Grosu(2016).
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