3 solved out of 3 shown
If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor n/2\rfloor$ distinct distances.
Solved by Altman [Al63]. The stronger variant that says there is one point which determines at least $\lfloor n/2\rfloor$ distinct distances is still open.
Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then $\sum_i f(u_i)^2 \ll n^3.$
Solved by Fishburn [Al63]. Note it is trivial that $\sum f(u_i)=\binom{n}{2}$. The stronger conjecture that $\sum f(u_i)^2$ is maximal for the regular $n$-gon (for large enough $n$) is still open.
$500 Let$x_1,\ldots,x_n\in\mathbb{R}^2$determine the set of distances$\{u_1,\ldots,u_t\}$. Suppose$u_i$appears as the distance between$f(u_i)$many pairs of points. Then for all$\epsilon>0$$\sum_i f(u_i)^2 \ll_\epsilon n^{3+\epsilon}.$ The case when the points determine a convex polygon was been solved by Fishburn [Al63]. Note it is trivial that$\sum f(u_i)=\binom{n}{2}\$. Solved by Guth and Katz [GuKa15] who proved the upper bound $\sum_i f(u_i)^2 \ll n^3\log n.$