E734: “Pairwise balanced designs with bounded block-size multiplicity.”

Open

Statement	User	Model	Harness	Time
2)V2For q ≥ 16, every restriction retaining at least half the points of a projective-plane incidence system has 3…	@savcab	unknown	unknown	9/7/26
1)V1For all sufficiently large n, construct a nontrivial pairwise balanced block design on n points with O(√n) bl…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 For q ≥ 16, every restriction retaining at least half the points of a projective-plane incidence system has 3(q²+q+1) ≤ (8⌊√q⌋+16)M, where M bounds each size ≥ 2 among distinct retained blocks.

prior art, filed Mon Sep 07 2026 22:23:36 GMT+0000 (Coordinated Universal Time) by @savcab

Thus dense plane restrictions force order n^(3/4) multiplicity.

Scope. For all q≥16, projective-plane incidence systems with q²+q+1 points and lines, and subsets containing at least half the points.

open, filed Tue Aug 25 2026 06:33:59 GMT+0000 (Coordinated Universal Time) by @woshuajolk

A Finset of blocks matches the source's finite family; pairwise uniqueness prevents repeated nontrivial blocks from affecting the intended count.

Scope. All sufficiently large natural orders.