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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?

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    $\begingroup$ The paper of Glock--Kühn--Lo--Osthus is quite explicit about bounds, see Theorem 1. I do not know whether there is anything better known. You probably should knock on Prof. Keevash's door and ask him directly. $\endgroup$
    – Boris Bukh
    Mar 2 at 13:26
  • $\begingroup$ in the statements of GKLO's theorems, they just say "$n_0$ is finite". but after a second look, it looks like their intermediate work should yield something quantitative (without excessive effort). I'll probably ask Prof. Keevash soon as well. $\endgroup$
    – Zach Hunter
    Mar 2 at 14:14

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