Erdős distinct distances problem

In discrete geometry, the Erdős distinct distances problem states that between n distinct points on a plane there are at least n^{1 − o(1)} distinct distances. It was posed by Paul Erdős in 1946 and proven by Guth & Katz (2015).

In what follows let g(n) denote the minimal number of distinct distances between n points in the plane. In his 1946 paper, Erdős proved the estimates

for some constant ${\displaystyle c}$. The lower bound was given by an easy argument. The upper bound is given by a ${\displaystyle {\sqrt {n}}\times {\sqrt {n}}}$ square grid. For such a grid, there are ${\displaystyle O(n/{\sqrt {\log n}})}$ numbers below n which are sums of two squares; see Landau–Ramanujan constant. Erdős conjectured that the upper bound was closer to the true value of g(n), and specifically that ${\displaystyle g(n)=\Omega (n^{c})}$ holds for every c < 1.

...
Wikipedia