Two Questions of Erdős on Hypergraphs above the Turán Threshold

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Abstract

For ordinary graphs it is known that any graph G with more edges than the Turán number of inline image must contain several copies of inline image, and a copy of inline image, the complete graph on inline image vertices with one missing edge. Erdős asked if the same result is true for inline image, the complete 3-uniform hypergraph on s vertices. In this note, we show that for small values of n, the number of vertices in G, the answer is negative for inline image. For the second property, that of containing a inline image, we show that for inline image the answer is negative for all large n as well, by proving that the Turán density of inline image is greater than that of inline image.

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