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Updated Oct 7

Problem 1159

Statement. 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 $\ell$ ?

Status. Open.

Source. erdosproblems.com/1159, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1159, https://www.erdosproblems.com/1159.

References.

[ESS83] Erdős, P. and Silverman, R. and Stein, A., Intersection properties of families containing sets of nearly the same size. Ars Combin. (1983), 247-259.
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (8)

Problem 664 Alon's partial design with no bounded blocking set Set Systems, Designs and Hypergraphs set_systems/erdos_1981_combinatorial_problems_which_i_would_most set_systems/erdos_1983_intersection_properties_families_containing_sets_nearly Corollary (p. 255): for every c > 2e, a projective plane of large order n has property B(c log n)Theorem 1 (p. 248): a family of at most n^b sets of sizes between a_1 n and a_2 n has a set S meeting each member in between c_1 n^delta log^s n and c_2 n^delta log^s n points Theorem 2 (p. 258): with k, k' as in Lemma 8 and j <= n/(2k'+1), a projective plane of order n has property B(n+2-j)

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