Problem Statement
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 $x\not\in A_y$ 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 $3$?
If the sets $A_x$ are closed and have measure $<1$, then must there exist an independent set of size $3$?
Categories:
Combinatorics Set Theory
Progress
Erdős and Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets (under the assumptions in the first problem).Gladysz [Gl62] proved the existence of an independent set of size $2$ under the assumptions of the the second question.
Hechler [He72] has shown the answer to the first question is no, assuming the continuum hypothesis.
Newelski, Pawlikowski, and Seredyński [NPS87] proved the answer to the first question is yes, under the additional assumption that the $A_x$ are closed.
Source: erdosproblems.com/501 | Last verified: January 15, 2026