A 1-avoiding set is a subset of $$\mathbb {R}^n$$ R n that does not contain pairs of points at distance 1. Let $$m_1(\mathbb {R}^n)$$ m 1 ( R n ) denote the maximum fraction of $$\mathbb {R}^n$$ R n that can be covered by a measurable 1-avoiding set. We prove two results. First, we show that any 1-avoiding set in $$\mathbb {R}^n (n\ge 2)$$ R n ( n ≥ 2 ) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than 1 and points from distinct blocks lie farther than 1 unit of distance apart from each other) has density strictly less than $$1/2^n$$ 1 / 2 n . For the special case of sets with block structure this proves a conjecture of Erdős asserting that $$m_1(\mathbb {R}^2) < 1/4$$ m 1 ( R 2 ) < 1 / 4 . Second, we use linear programming and harmonic analysis to show that $$m_1(\mathbb {R}^2) \le 0.258795$$ m 1 ( R 2 ) ≤ 0.258795 .