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Claim. The question of Problem 732, as its precise Statement poses it, is answered yes. Theorem 2.3 of Noga Alon's note Blocking partial designs and block-compatible sequences, published as Theorem 4.5 of Section 4 of Alon's chapter Problems and Results in Extremal Combinatorics–V (card; the chapter's Problem 4.2 is Problem 732), states: for a large prime power $q$ and $n=q^2+q+1$, every nonincreasing integer sequence $q+1\ge x_1\ge\cdots\ge x_n\ge3$, followed by $\sum_{i\le n}\left[\binom{q+1}2-\binom{x_i}2\right]$ entries equal to $2$, is block-compatible for $n$; hence the number of block-compatible sequences for $n$ is at least

$$\binom{n+q-2}{q-2}=2^{(1/2+o(1))\,n^{1/2}\log n}$$

with logarithms to base $2$. The design is built from the lines of a projective plane of order $q$: each line is shrunk to a subset of size $x_i$, and every pair of points of the line left uncovered becomes a block of size $2$. The note states the bound for $n$ of the form $q^2+q+1$. Two observations, made here, carry it to every large $n$: adding a point to an $n$-point design together with the $n$ blocks of size $2$ joining it to the old points gives a design on $n+1$ points, so the number of block-compatible sequences does not decrease with $n$; and Bertrand's postulate gives a prime $q$ with $q^2+q+1\le n$ and $q\gg n^{1/2}$. So at least $\exp(cn^{1/2}\log n)$ sequences are block-compatible for every large $n$, for an absolute $c>0$. Section 3 of the note, the Remarks of Section 4.3 of the chapter, also proves Erdős's upper bound $2^{O(n^{1/2}\log n)}$, so the count is $\exp(\Theta(n^{1/2}\log n))$. The note writes a sequence in nonincreasing order with $x_m\ge2$; the site's $1<X_1\le\cdots\le X_m\le n$ is the same sequence reversed.

Covers. The whole precise Statement: for an absolute $c>0$ and every large $n$, at least $\exp(cn^{1/2}\log n)$ sequences are block-compatible for $\{1,\ldots,n\}$. Theorem 2.3 of the note, Theorem 4.5 of the chapter, is stated for $n=q^2+q+1$ only; the passage to every large $n$ is the two observations above, monotonicity in $n$ by adding a point with $n$ blocks of size $2$, and Bertrand's postulate. Alon leaves that passage implicit while presenting Theorem 4.5 as settling the chapter's Problem 4.2 for all large $n$.

Acceptance. Reviewed: the site's curator, Thomas Bloom, labels the problem proved and credits the lower bound to Alon, linking the note, independently of its author. The note carries no date and is listed without a venue on the author's publications page; the page is dated by the server's modification date of the file, 2024-08-03, the only date available, and the note's references reach to 2020. The result was then published as Theorem 4.5 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 the page lists no refereed evidence. The chapter's proof of Theorem 4.5 takes the lines of a projective plane of order $q$, correcting the "order $q+1$" of the note's first sentence. A Lean 4 development by Jingxuan Ding of the SpringSense Innovation Institute, posted on the site's discussion thread on 2026-10-04 and pinned above at its commit, names Alon's Problem 1.2 and Theorem 2.3 as its source and proves erdos732_yes: there is $c>0$ such that for all large $n$ at least $\exp(c\sqrt n\ln n)$ nonincreasing lists with entries in $[2,n]$ are realized as the block sizes of a pairwise balanced design on $n$ points. Its statement audit records the passage to every large $n$ through monotonicity and Bertrand's postulate, and notes that the first sentence of the note's proof of Theorem 2.3 says a plane of order $q+1$ where the theorem needs order $q$. The development's README says its paper analysis was AI-assisted and names Codex for release packaging. This corpus has not built or audited it, so the page lists no formalized evidence.