  A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs
of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction
of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We
prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$
($n\ge 2$) 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 $1/2^n$. For the special case of
sets with block structure this proves a conjecture of Erd\H{o}s asserting that
$m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic
analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$.
