I am studying the paper DIOPHANTINE EQUATIONS OF THE FORM $F(X) = G(Y)$ - AN EXPOSITION which discusses the result of Erdos and Selfridge.
I am unable to understand the highlighted statement ''Clearly each prime factor of each $a_i$ is less than $k$."
For example, if I take $n=49$ and $k=4$, then we have $50\cdot51\cdot52\cdot53 = y^2$ and $$50 = 2 \cdot 5^2, \quad \quad 51 = 3\cdot17, \quad \quad 52= 13\cdot2^2, \quad \quad 53= 53.$$
Clearly, $13 \nleq 4.$ What am I misunderstanding here? Please help me.
The highlighted statement only applies if $r:=\prod_{i=1}^k a_i$ is a square, which is not the case in your example.
Let $p$ be prime factor of $a_i$ with $p \ge k$. Then $p$ divides $n+i$, but does not divide any other of the factors $n+1,..n+k$. Therefore, $p^2$ does not divide $r$, and $r$ is not a square, in contradiction to the assumption.