The board / combinatorics / #501
Erdős Problem #501
NOT DISPROVABLEdatabase updated 2026-09-03
Open in general, but there are models of set theory in which it is true — so it can't be disproved from the usual axioms (ZFC) alone.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
For every \(x \in \mathbb{R}\) let \(A_x \subset \mathbb{R}\) be a bounded set with outer measure \(< 1\). Must there exist an infinite independent set, that is, some infinite \(X \subseteq \mathbb{R}\) such that \(x \notin A_y\) for all \(x \neq y \in X\)? If the sets \(A_x\) are closed and have measure \(< 1\), then must there exist an independent set of size \(3\)? Known results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets. Hechler [He72] showed the answer is **no** assuming the continuum hypothesis.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/501.Prizenone
Topicscombinatorics, set theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/501
Attempts
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A null-initial well-order gives an explicit counterexample, so the first question is negative under add(N)=cov(N) or non(N)=c; the live joint problem remains OPEN.
PARTIAL