On A Hypergraph Turán Problem Of Frankl

Let \( C^{{{\left( {2k} \right)}}}_{r} \) be the 2k-uniform hypergraph obtained by letting P1, . . .,Pr be pairwise disjoint sets of size k and taking as edges all sets P i P j with ij. This can be thought of as the ‘k-expansion’ of the complete graph K r : each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain \( C^{{{\left( {2k} \right)}}}_{3} \) can be obtained by partitioning V = V1 ∪V2 and taking as edges all sets of size 2k that intersect each of V1 and V2 in an odd number of elements. Let \( {\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n} \) denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no \( C^{{{\left( {2k} \right)}}}_{3} \) has at most as many edges as \( {\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n} \).

Sidorenko has given an upper bound of \( \frac{{r - 2}} {{r - 1}} \) for the Tur´an density of \( C^{{{\left( {2k} \right)}}}_{r} \) for any r, and a construction establishing a matching lower bound when r is of the form 2p+1. In this paper we also show that when r=2p+1, any \( C^{{{\left( 4 \right)}}}_{r} \)-free hypergraph of density \( \frac{{r - 2}} {{r - 1}} - o{\left( 1 \right)} \) looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of \( C^{{{\left( 4 \right)}}}_{r} \) to \( \frac{{r - 2}} {{r - 1}} - c{\left( r \right)} \), where c(r) is a constant depending only on r.

The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials.

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Correspondence to Peter Keevash.

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* Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.

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Keevash, P., Sudakov*, B. On A Hypergraph Turán Problem Of Frankl. Combinatorica 25, 673–706 (2005). https://doi.org/10.1007/s00493-005-0042-2

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Mathematics Subject Classification (2000):

  • 05C35
  • 05C65
  • 05D05
  • 05E35