  For positive integers $n>d\geq k$, let $\phi(n,d,k)$ denote the least integer
$\phi$ such that every $n$-vertex graph with at least $\phi$ vertices of degree
at least $d$ contains a path on $k+1$ vertices. Many years ago, Erd\H{o}s,
Faudree, Schelp and Simonovits proposed the study of the function
$\phi(n,d,k)$, and conjectured that for any positive integers $n>d\geq k$, it
holds that $\phi(n,d,k)\leq
\lfloor\frac{k-1}{2}\rfloor\lfloor\frac{n}{d+1}\rfloor+\epsilon$, where
$\epsilon=1$ if $k$ is odd and $\epsilon=2$ otherwise. In this paper we
determine the value of the function $\phi(n,d,k)$ exactly. This confirms the
above conjecture of Erd\H{o}s et al. for all positive integers $k\neq 4$ and in
a corrected form for the case $k=4$. Our proof utilizes, among others, a lemma
of Erd\H{o}s et al., a theorem of Jackson, and a (slight) extension of a very
recent theorem of Kostochka, Luo and Zirlin, where the latter two results
concern maximum cycles in bipartite graphs. Besides, we construct examples to
provide answers to two closely related questions raised by Erd\H{o}s et al.
