LetC(3)ndenote the 3-uniformtight cycle, that is, the hypergraph with verticesv1, .–.–.,vnand edgesv1v2v3,v2v3v4, .–.–.,vn−1vnv1,vnv1v2. We prove that the smallest integerN=N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph withNvertices contains a monochromatic copy ofC(3)nis asymptotically equal to 4n/3 ifnis divisible by 3, and 2notherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.
Let r 2 be an integer. The real number α ∈ [0, 1] is a jump for r if there exists c > 0 such that for every positive and every integer m r, every r-uniform graph with n > n 0 ( , m) vertices and at least (α + ) n r edges contains a subgraph with m vertices and at least (α + c) m r edges. A result of Erdős, Stone and Simonovits implies that every α ∈ [0, 1) is a jump for r = 2. For r 3, Erdős asked whether the same is true and showed that every α ∈ [0, r! r r ) is a jump. Frankl and Rödl gave a negative answer by showing that 1 − 1 l r−1 is not a jump for r if r 3 and l > 2r. Another well-known question of Erdős is whether r! r r is a jump for r 3 and what is the smallest non-jumping number. In this paper we prove that 5 2 r! r r is not a jump for r 3. We also describe an infinite sequence of non-jumping numbers for r = 3.
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