Abstract: Extremal problems for hypergraphs concern the maximum density of large hypergraphs H that do not contain a copy of a given hypergraph F. Estimating the so-called Turán-densities is a central problem in combinatorics. However, despite a lot of effort precise estimates are know for only known for very few hypergraphs F.

We consider a variation of the problem, where the large hypergraphs H satisfy in addition quasirandom conditions on its edge distribution. We present recent progress based on joint work with Chr. Reiher and V. Rödl. In particular, we established a computer free proof of a recent result of Glebov, Kral and Volec on the Turán-density on weakly quasirandom hypergraphs not containing the 3-uniform hypergraph with three edges on four vertices.

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