Erdős Problem #1159

Determine whether there exists a constant $C>1$ such that the following holds.

Let $P$ be a finite projective plane. Must there exist a set of points $S$ such that $1\leq \lvert S\cap \ell\rvert \leq C$ for all lines $\ell$?

#1159: [Va99,4.70]

combinatorics

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A set which meets all lines at once is called a blocking set. In [Er81] Erdős asks the stronger question of whether this is true for all pairwise balanced block designs.

Erdős, Silverman, and Stein [ESS83] proved this is true with $\lvert S\cap\ell \rvert\ll \log n$ for all lines $\ell$ (where $n$ is the order of the projective plane).

See also [664] for a stronger question.

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This page was last edited 23 January 2026.

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