  The generalized Ramsey number $f(n, p, q)$ is the smallest number of colors
needed to color the edges of the complete graph $K_n$ so that every $p$-clique
spans at least $q$ colors. Erd\H{o}s and Gy\'arf\'as showed that $f(n, p, q)$
grows linearly in $n$ when $p$ is fixed and $q=q_{\text{lin}}(p):=\binom
p2-p+3$. Similarly they showed that $f(n, p, q)$ is quadratic in $n$ when $p$
is fixed and $q=q_{\text{quad}}(p):=\binom p2-\frac p2+2$. In this note we
improve on the known estimates for $f(n, p, q_{\text{lin}})$ and $f(n, p,
q_{\text{quad}})$. Our proofs involve establishing a significant strengthening
of a previously known connection between $f(n, p, q)$ and another extremal
problem first studied by Brown, Erd\H{o}s and S\'os, as well as building on
some recent progress on this extremal problem by Delcourt and Postle and by
Shangguan. Also, our upper bound on $f(n, p, q_{\text{lin}})$ follows from an
application of the recent forbidden submatchings method of Delcourt and Postle.
