Disproved on the site (label DISPROVED; page last edited 27 January 2026). The site remarks that a positive answer would give every finite geometry a blocking set meeting each line in a bounded number of points, that Erdős's formulation in [Er81] asks instead about pairwise balanced block designs (every pair of points in exactly one $A_i$, with the size condition $|A_i|>c\sqrt n$ kept for every $c>0$ and no restriction $c<1$), and that Alon answered the question no: for a large prime power $q$ and $n=m=q^2+q+1$ there are sets with $|A_i|\ge\tfrac25\sqrt n$ and pairwise intersections of size at most $1$ such that every $B$ meeting all of them meets some $A_j$ in $\gg\log n$ points, each set a random half of a line of a projective plane of order $q$. The standing rests on Alon's construction, accepted on the curator's credit; the note was published in 2026 as Section 4 of Alon's chapter Problems and Results in Extremal Combinatorics–V (card), in a proceedings volume. The block-design version of [Er81] remains open, with Alon conjecturing a negative answer there too, and Problem 1159 asks the projective-plane case.