Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a hypergraph Turan problem of Frankl   Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting
$P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all
sets $P_i \cup P_j$ with $i \neq j$. 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 $k$. We determine the exact Turan number of $C^{2k}_3$ and the
corresponding extremal hypergraph, thus confirming a conjecture of Frankl.
Sidorenko has given an upper bound of $(r-2) / (r-1)$ for the Tur\'an density
of $C^{2k}_r$ for any $r$, and a construction establishing a matching lower
bound when $r$ is of the form $2^p + 1$. We show that when $r = 2^p + 1$, any
$C^4_r$-free hypergraph of density $(r-2)/(r-1) - o(1)$ 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 Tur\'an density of $C^4_r$ to $(r-2)/(r-1) - c(r)$, 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.

11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
3
The categories of this paper are: physics
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The categories of this paper are: comp-gas