  Recently, a relation between Schreier-type sets and Tur\'{a}n graphs was
discovered. In this note, we give a combinatorial proof and obtain a
generalization of the relation. Specifically, for $p, q\ge 1$, let
$$\mathcal{A}_q := \{F\subset\mathbb{N}: |F| = 1 \mbox{ or }F\mbox{ is an
arithmetic progression with difference } q\}$$ and $$Sr(n, p, q)\ :=\
\#\{F\subset \{1, \ldots, n\}\,:\, p\min F\ge |F|\mbox{ and }F\in
\mathcal{A}_q\}.$$ We show that $$Sr(n, p, q) \ =\ T(n+1, pq+1, q),$$ where
$T(\cdot, \cdot, \cdot)$ is the number of edges of an $n$-vertex graph that is
a modification of Tur\'{a}n graphs. We also prove that $Sr(n,p,q)$ is the
partial sum of certain sequences.
