For $A\subset {\mathbb{R}}^2$ we define the upper density as$$\overline{\delta }(A)=\underset{R\to \infty }{\mathrm{limsup}}\frac{\lambda (A\cap B_R)}{\lambda (B_R)},$$where $\lambda$ is the Lebesgue measure and $B_R$ is the ball of radius $R$.Estimate$$m_1=\sup \overline{\delta }(A),$$where $A$ ranges over all measurable subsets of ${\mathbb{R}}^2$ without two points distance $1$ apart. In particular, is $m_1\le 1/4$?