Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the algebraic and topological structure of the set of Tur\'an
  densities   The present paper is concerned with the various algebraic structures
supported by the set of Tur\'an densities.
  We prove that the set of Tur\'an densities of finite families of r-graphs is
a non-trivial commutative semigroup, and as a consequence we construct explicit
irrational densities for any r >= 3. The proof relies on a technique recently
developed by Pikhurko.
  We also show that the set of all Tur\'an densities forms a graded ring, and
from this we obtain a short proof of a theorem of Peng on jumps of hypergraphs.
  Finally, we prove that the set of Tur\'an densities of families of r-graphs
has positive Lebesgue measure if and only if it contains an open interval. This
is a simple consequence of Steinhaus's theorem.

11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph