Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Induced subgraphs with many distinct degrees   Let $\hom(G)$ denote the size of the largest clique or independent set of a
graph $G$. In 2007, Bukh and Sudakov proved that every $n$-vertex graph $G$
with $\hom(G) = O(\log n)$ contains an induced subgraph with $\Omega(n^{1/2})$
distinct degrees, and raised the question of deciding whether an analogous
result holds for every $n$-vertex graph $G$ with $\hom(G) = O(n^\epsilon)$,
where $\epsilon > 0$ is a fixed constant. Here, we answer their question in the
affirmative and show that every graph $G$ on $n$ vertices contains an induced
subgraph with $\Omega((n/\hom(G))^{1/2})$ distinct degrees. We also prove a
stronger result for graphs with large cliques or independent sets and show, for
any fixed $k \in \mathbb{N}$, that if an $n$-vertex graph $G$ contains no
induced subgraph with $k$ distinct degrees, then $\hom(G) \ge n/(k-1)-o(n)$;
this bound is essentially best-possible.

37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph