Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sets without $k$-term progressions can have many shorter progressions   Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic
progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains
no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove
that $$\lim_{n \to \infty} \frac{\log f_{s,k}(n)}{\log n} = 2,$$ which answers
an old question of Erd\H{o}s. In fact, we prove upper and lower bounds for
$f_{s,k}(n)$ which show that its growth is closely related to the bounds in
Szemer\'edi's theorem.

20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat