Near-uniform designs and size multiplicity

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Problem 665 asks for pairwise balanced designs whose blocks are almost of square-root size, whereas Problem 734 asks that no size occur too often. The two requirements do not compose through near-uniformity. The complete author-recorded obstruction is L26: a nontrivial design has at least $n$ blocks; minimum size $(1-o(1))\sqrt n$ confines all its sizes to an interval containing only $o(\sqrt n)$ integers. A fixed bound of $C\sqrt n$ blocks of each size would then allow only $o(n)$ blocks.

The constructive proof of Erdős–Larson Theorem 1 uses at most six sizes, so some size occurs at least $n/6$ times. More generally, a successful construction for Problem 734 must use at least $\sqrt n/C$ distinct sizes. This is a necessary condition, not a construction or a disproof of the problem.

The source's nearby-prime construction has a separate limitation for Problem 665. Starting with a plane of prime order $p$, write the excess $p^2+p+1-n=rp+1+s$, $0\le s<p$, when it is nonzero. Deleting a point, $r$ suitable lines through it, and $s$ conic points leaves block sizes at least $\min(p-2,p-r-1)\ge p-r-2$. Since $rp\le p^2+p-n$, this is at least $n/p-3$. For $p\ge\sqrt n$ the deficit from $\sqrt n$ is at most $p-\sqrt n+3$; for $p<\sqrt n$ the bound exceeds $\sqrt n-3$. At zero excess the plane itself suffices. The construction therefore transfers the prime gap into the block deficit; it does not make the deficit bounded by an absolute constant. Its geometric construction was inspected as an author source check. The converse embedding result cited on Problem 665 was read only through its existing account and is not a checked premise here.

A useful new route must construct many sizes while keeping each multiplicity small, or replace the nearby-prime reduction with an all-order construction having a bounded deficit. The present obstruction only excludes the near-uniform route to the multiplicity target.