Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the number of 4-cycles in a tournament   If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what
can be said about the number of its $4$-cycles? The most interesting range of
this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for
some $c>0$ and we seek to minimize the number of $4$-cycles. We conjecture that
the (asymptotic) minimizing $T$ is a random blow-up of a constant-sized
transitive tournament. Using the method of flag algebras, we derive a lower
bound that almost matches the conjectured value. We are able to answer the
easier problem of maximizing the number of $4$-cycles. These questions can be
equivalently stated in terms of transitive subtournaments. Namely, given the
number of transitive triples in $T$, how many transitive quadruples can it
have? As far as we know, this is the first study of inducibility in
tournaments.

22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th