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OPEN - $500 Does every set of$n$distinct points in$\mathbb{R}^2$determine$\gg n/\sqrt{\log n}$many distinct distances? A$\sqrt{n}\times\sqrt{n}$integer grid shows that this would be the best possible. Nearly solved by Guth and Katz [GuKa15] who proved that there are always$\gg n/\log n$many distinct distances. A stronger form (see [604]) may be true: is there a single point which determines$\gg n/\sqrt{\log n}$distinct distances, or even$\gg n$many such points, or even that this is true averaged over all points. See also [661]. OPEN -$500
Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart?
The unit distance problem. In [Er94b] Erdős dates this conjecture to 1946.

This would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(n^{4/3})$, due to Spencer, Szemerédi, and Trotter [SST84]. In [Er83c] and [Er85] Erdős offers \$250 for an upper bound of the form$n^{1+o(1)}$. Part of the difficulty of this problem is explained by a result of Valtr (see [Sz16]), who constructed a metric on$\mathbb{R}^2$and a set of$n$points with$\gg n^{4/3}$unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemerédi, and Trotter [SST84] generalise to include this metric. Therefore to prove an upper bound better than$n^{4/3}$some special feature of the Euclidean metric must be exploited. See a survey by Szemerédi [Sz16] for further background and related results. See also [92], [96], and [605]. SOLVED Let$f(n)$be minimal such that the following holds. For any$n$points in$\mathbb{R}^2$, not all on a line, there must be at least$f(n)$many lines which contain exactly 2 points (called 'ordinary lines'). Does$f(n)\to \infty$? How fast? Conjectured by Erdős and de Bruijn. The Sylvester-Gallai theorem states that$f(n)\geq 1$. The fact that$f(n)\geq 1$was conjectured by Sylvester in 1893. Erdős rediscovered this conjecture in 1933 and told it to Gallai who proved it. That$f(n)\to \infty$was proved by Motzkin [Mo51]. Kelly and Moser [KeMo58] proved that$f(n)\geq\tfrac{3}{7}n$for all$n$. This is best possible for$n=7$. Motzkin conjectured that for$n\geq 13$there are at least$n/2$such lines. Csima and Sawyer [CsSa93] proved a lower bound of$f(n)\geq \tfrac{6}{13}n$when$n\geq 8$. Green and Tao [GrTa13] proved that$f(n)\geq n/2$for sufficiently large$n$. (A proof that$f(n)\geq n/2$for large$n$was earlier claimed by Hansen but this proof was flawed.) The bound of$n/2$is best possible for even$n$, since one could take$n/2$points on a circle and$n/2$points at infinity. Surprisingly, Green and Tao [GrTa13] show that if$n$is odd then$f(n)\geq 3\lfloor n/4\rfloor$. SOLVED For$A\subset \mathbb{R}^2$we define the upper density as $\overline{\delta}(A)=\limsup_{R\to \infty}\frac{\lambda(A \cap B_R)}{\lambda(B_R)},$ 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$m_1\leq 1/4$? A question of Moser [Mo66]. A lower bound of$m_1\geq \pi/8\sqrt{3}\approx 0.2267$is given by taking the union of open circular discus of radius$1/2$at a regular hexagonal lattice suitably spaced aprt. Croft [Cr67] gives a small improvement of$m_1\geq 0.22936$. The trivial upper bound is$m_1\leq 1/2$, since for any unit vector$u$the sets$A$and$A+u$must be disjoint. Erdős' question was solved by Ambrus, Csiszárik, Matolcsi, Varga, and Zsámboki [ACMVZ23] who proved that$m_1\leq 0.247$. OPEN -$500
Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that $\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?$ Or even $\gg n/\sqrt{\log n}$?
The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\sqrt{\log n}$ would be best possible.

It may be true that there are $\gg n$ many such points, or that this is true on average. In [Er97e] Erdős offers \$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for$\gg n$many such points. In [Er97e] Erdős wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of$n^{o(1)}$. The best known bound is $\gg n^{c-o(1)},$ due to Katz and Tardos [KaTa04], where $c=\frac{48-14e}{55-16e}=0.864137\cdots.$ SOLVED Is there some function$f(n)\to \infty$as$n\to\infty$such that there exist$n$distinct points on the surface of a two-dimensional sphere with at least$f(n)n$many pairs of points whose distances are the same? See also [90]. This was solved by Erdős, Hickerson, and Pach [EHP89]. For$D>1$and$n\geq 2$let$u_D(n)$be such that there is a set of$n$points on the sphere in$\mathbb{R}^3$with radius$D$such that there are$u_D(n)$many pairs which are distance$1$apart (so that this problem asked for$u_D(n)\geq f(n)n$for some$D$). Erdős, Hickerson, and Pach [EHP89] proved that$u_{\sqrt{2}}(n)\asymp n^{4/3}$and$u_D(n)\gg n\log^*n$for all$D>1$and$n\geq 2$(where$\log^*$is the iterated logarithm function). This lower bound was improved by Swanepoel and Valtr [SwVa04] to$u_D(n) \gg n\sqrt{\log n}$. The best upper bound for general$D$is$u_D(n)\ll n^{4/3}$. Additional thanks to: Dmitrii Zakharov SOLVED Given any$n$distinct points in$\mathbb{R}^2$let$f(n)$count the number of distinct lines determined by these points. What are the possible values of$f(n)$? A question of Grünbaum. The Sylvester-Gallai theorem implies that if$f(m)>1$then$f(m)\geq n$. Erdős showed that, for some constant$c>0$, all integers in$[cn^{3/2},\binom{n}{2}]$are possible except$\binom{n}{2}-1$and$\binom{n}{2}-3$. Solved (for all sufficiently large$n$) completely by Erdős and Salamon [ErSa88]; the full description is too complicated to be given here. OPEN -$250
For a set of $n$ points $P\subset \mathbb{R}^2$ let $\ell_1,\ldots,\ell_m$ be the lines determined by $P$, and let $A=\{\lvert \ell_1\cap P\rvert,\ldots,\lvert \ell_m\cap P\rvert\}$.

Let $F(n)$ count the number of possible sets $A$ that can be constructed this way. Is it true that $F(n) \leq \exp(O(\sqrt{n}))?$

Erdős writes it is 'easy to see' that this bound would be best possible.