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Two Questions of Erdős on Hypergraphs above the Turán Threshold



For ordinary graphs it is known that any graph G with more edges than the Turán number of math formula must contain several copies of math formula, and a copy of math formula, the complete graph on math formula vertices with one missing edge. Erdős asked if the same result is true for math formula, 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 math formula. For the second property, that of containing a math formula, we show that for math formula the answer is negative for all large n as well, by proving that the Turán density of math formula is greater than that of math formula.