Given $s>t\ge 2$, let $N(s,t)$ be the smallest integer $n>s$ such that there exists an “$(n;s;t;1)$-design” (i.e., a collection of $s$-subsets $e_1,\dots,e_m$ of $[n]:=\{1,\dots,n\}$, such that every $t$-subset of $[n]$ is contained by exactly one $e_i$ in our collection). A celebrated result of Keevash tells us that $N(s,t)$ is always finite.
Already for $t=2$, I hear this function is not fully understood (it is related to understanding when projective planes exist). It is known that if $s-1$ is of the form $p^k$ for some prime $p$, then $N(s,2)= s^2-s+1$. Conversely, it is conjectured that $N(s,2)>s^2-s-1$ for all other $s$; but this is still open for a positive fraction of integers $s$…
Question: What upper bounds are known for $N(s,t)$? Is there any good references on this topic?
