Editing Openai/68ec50da-cf00-8005-b5f6-b683506e5853 (section)

==== Previously I used the fact that for a single aaa a positive fraction (δ=∏p≠5(1−1/p2)=25/(4π2)\delta=\prod_{p\ne5}(1-1/p^2)=25/(4\pi^2)δ=∏p=5(1−1/p2)=25/(4π2)) of b∈Stb\in S_tb∈St give ab+1ab+1ab+1 squarefree. That does not by itself control how AsA_sAs could be arranged adversarially (it might live entirely inside the “bad” set for that particular aaa). ====

The new Lemma 2.1 and Corollary 2.2 handle the simultaneous constraints imposed by many different aaa’s: every time we add another aia_iai, the set of bbb that can still satisfy “∃p∈P\exists p\in P∃p∈P with p2∣aib+1p^2\mid a_i b+1p2∣aib+1” shrinks by a multiplicative factor ≤SP\le S_P≤SP, uniformly and without independence assumptions. Since SP<1S_P<1SP<1 and we can take k→∞k\to\inftyk→∞ if one 25‑class contains ≫N\gg N≫N elements, the intersection becomes negligible, forcing all other residue classes to be o(N)o(N)o(N).

===== The inclusion–exclusion you had in mind does yield the full extremal result once we organize it via the uniform covering inequality (2.2): it shows that the only asymptotically extremal sets are a single 252525-class r∈{7,18}r\in\{7,18\}r∈{7,18}, and hence the sharp upper bound is ∣A∣≤N/25+o(N)|A|\le N/25+o(N)∣A∣≤N/25+o(N). =====

If you’d like, I can also write the same argument with explicit, fully effective constants (choose a concrete PPP, say P={2,3,7,11,13,…,P0}P=\{2,3,7,11,13,\dots, P_0\}P={2,3,7,11,13,…,P0}, compute SP,TPS_P,T_PSP,TP and the exact kkk needed); but the qualitative stability—“one 25‑class survives, all others are o(N)o(N)o(N)”—is already proved above.