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Claim. The answer is no. Noga Alon's note Blocking partial designs and block-compatible sequences, published as Section 4 of Alon's chapter Problems and Results in Extremal Combinatorics–V (card), states the question as Problem 1.1 (the chapter's Problem 4.1, which cites problem number 664): whether for every fixed constant $c<1$ there is $C=C(c)$ such that any sets $A_1,\dots,A_m\subseteq\{1,\dots,n\}$ with $|A_i|>c\sqrt n$ and $|A_i\cap A_j|\le1$ for $i\ne j$ admit a set $B$ with $0<|B\cap A_i|\le C$ for every $i$. Its Theorem 2.1 (the chapter's Theorem 4.3) refutes this: for a large prime power $q$ and $m=n=q^2+q+1$ there are $m$ subsets of an $n$-element set, each of size more than $0.4\sqrt n$, pairwise meeting in at most one point, such that every $B$ meeting all of them satisfies $|B\cap A_i|\ge0.1\log_2 n$ for some $i$. The construction takes the $m$ lines of a projective plane of order $q$ and keeps each point of each line independently with probability $1/2$; with high probability the pieces have the stated sizes and intersections, and a counting argument over the sets $B$ of size at most $0.3\sqrt n\log_2 n$ shows that none of them meets every piece, so a blocking set has more than $0.3\sqrt n\log_2 n$ points and, by averaging over the roughly $\sqrt n/2$ pieces through each point, meets some piece in more than $0.1\log_2 n$ of them. Since $|A_i|>0.4\sqrt n$ implies $|A_i|>c\sqrt n$ for every $c\le2/5$, the question of Problem 664, asked for every $c<1$, has a negative answer. The note's Proposition 3.1 (the chapter's Proposition 4.6) shows the bound is sharp up to constants: when every block has size between $c_1\sqrt n$ and $c_2\sqrt n$, a random $B$ meets every block in $\Theta(\log n)$ points.

What remains open. The negative answer is proved at $c=2/5$, hence for every $c\le2/5$, which settles the question as asked, since it is posed for every fixed $c<1$; the note does not treat other values of $c$. Whether the full lines of a projective plane admit a blocking set meeting every line in a bounded number of points is Problem 1159, which is open. The version Erdős posed in [Er81], in which every pair of points lies in exactly one $A_i$ (a pairwise balanced block design), with the size condition $|A_i|>c\sqrt n$ kept for every $c>0$ and no restriction $c<1$, is not settled by the construction, whose pieces cover only some pairs; the note conjectures that the answer there is also no (its Conjecture 3.2, the chapter's Conjecture 4.7, on random subsets of the points of a projective plane) and records that the container-method results of Balogh–Samotij and Balogh–Solymosi have different parameters. Theorem 2.3 of the same note concerns block-compatible sequences and bears on Problem 732, not on this problem.

Acceptance. Reviewed: Thomas Bloom, the site's curator, marks the problem disproved and credits the construction to Alon, restating its parameters ($|A_i|\ge\tfrac25\sqrt n$, $|B\cap A_j|\gg\log n$). The note is posted on the author's page at Princeton; its statements and proofs were then published, with the same wording, as Section 4 of Noga Alon, Problems and Results in Extremal Combinatorics–V, in Sum(m)it280, Bolyai Society Mathematical Studies 32, Springer (2026), 13–29, online 2026-05-28, a proceedings volume whose refereeing is not documented, so no refereed evidence is listed. The note prints no date; the file's own creation date and the server's date for it are both 3 August 2024, which dates the page. No independent review of the proof is recorded.

Formalization. Boris Alexeev's repository holds a Lean 4 development, added on 2026-08-17, whose header calls it a formalization of a solution to the problem and names Alon as the informal author and "Codex" and "GPT-5.6 Sol" as the formal authors. Its top-level theorem is the $c=2/5$ case: there is no $K$ such that every family with $|A_i|>\tfrac25\sqrt n$ and pairwise intersections of size at most $1$ has a set $B$ meeting every member with $|B\cap A_i|\le K$ for all $i$; it follows from a theorem supplying a counterexample against every proposed bound $K$. The development takes the affine part of a Desarguesian projective plane in place of the full plane and credits the random half-line construction to Alon, Kalai, Matoušek and Meshulam; the file is linked above at the commit the formal-conjectures statement file pins when it names the development as the problem's formal proof. This corpus has not built or audited that development, so the page lists no formalized evidence; the acceptance rests on the curator's credit.