Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete subgraphs in a multipartite graph   In 1975 Bollob\'as, Erd\H os, and Szemer\'edi asked the following question:
given positive integers $n, t, r$ with $2\le t\le r-1$, what is the largest
minimum degree $\delta(G)$ among all $r$-partite graphs $G$ with parts of size
$n$ and which do not contain a copy of $K_{t+1}$? The $r=t+1$ case has
attracted a lot of attention and was fully resolved by Haxell and Szab\'{o},
and Szab\'{o} and Tardos in 2006. In this paper we investigate the $r>t+1$ case
of the problem, which has remained dormant for over forty years. We resolve the
problem exactly in the case when $r \equiv -1 \pmod{t}$, and up to an additive
constant for many other cases, including when $r \geq (3t-1)(t-1)$. Our
approach utilizes a connection to the related problem of determining the
maximum of the minimum degrees among the family of balanced $r$-partite
$rn$-vertex graphs of chromatic number at most $t$.

12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys