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For d geq 2 let F d(n) be minimal such that every set of n points in mathbb R ^d contains a set of F d(n) points with distinct distances. Estimate F d(n) for fixed d as n to infty .
Let P d(n) be such that in any set of n points in mathbb R ^d there exist at least P d(n) many points which do not contain an isosceles triangle. Estimate P d(n) - in particular, is it true that P 2(n)<n^ …
Let f d(n) be the minimal m such that any set of m points in mathbb R ^d contains a set of n points such that any two determined distances are distinct. Estimate f d(n) . In particular, is it true that, for f…
Let f(n) be minimal such that every set of n points in mathbb R ^2 contains at most f(n) many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate f(n) -…
Let g(n) be minimal such that any set of n points in mathbb R ^2 contains the vertices of at most g(n) many triangles with the same area. Estimate g(n) .
Let f d(n) be minimal such that, in any set of n points in mathbb R ^d , there exist at most f d(n) pairs of points which distance 1 apart. Estimate f d(n) .
Let f d(n) be minimal such that in any collection of n points in mathbb R ^d , all of distance at least 1 apart, there are at most f d(n) many pairs of points which are distance 1 apart. Estimate f d(n) .
Let d geq 3 , and let f d(n) be the minimal m such that every set of n points in mathbb R ^d determines at least m distinct distances. Estimate f d(n) - in particular, is it true that f d(n)=n^ frac 2 …
Let A subset mathbb R ^2 be a set of n points with no three on a line. Does A determine at least lfloor n/2 rfloor distinct distances? In fact, must there exist a single point from which there are at least $ lf…
Let f(n) be maximal such that, given any n points in mathbb R ^2 , there exist f(n) points such that no two are distance 1 apart. Estimate f(n) . In particular, is it true that f(n) geq n/4 ?
If n distinct points in mathbb R ^2 form a convex polygon then some vertex has at least lfloor frac n 2 rfloor different distances to other vertices.
Let A subset mathbb R ^2 be a set of size n and let d 1, ldots,d k be the set of distinct distances determined by A . Let f(d) be the number of times the distance d is determined, and suppose the d i ar…
If C,D subseteq mathbb R ^2 then the distance between C and D is defined by delta(C,D)= inf substack c in C d in D | c-d |. Let h(n) be the maximal number of unit distances between disjoint convex…
Let A subset x in mathbb R ^2 : lvert x rvert <r be a measurable set with no integer distances, that is, such that lvert a-b rvert not in mathbb Z for any distinct a,b in A . How large can the measure of …
Let f(n) be maximal such that any n points in mathbb R ^2 , with no three on a line, determine at least f(n) different convex subsets. Estimate f(n) - in particular, does there exist a constant c such that $…
Let h(n) be maximal such that in any n points in mathbb R ^2 (with no three on a line and no four on a circle) there are at least h(n) many circles of different radii passing through three points. Estimate $h(n)…
Let n k be minimal such that if n k points in mathbb R ^2 are in general position then there exists a subset of k points such that all binom k 3 triples determine circles of different radii. Determine $n k…
Let c(n) be minimal such that if k geq c(n) then the n -dimensional unit cube can be decomposed into k homothetic n -dimensional cubes. Give good bounds for c(n) - in particular, is it true that c(n) gg n^n ?
Let A subset mathbb R be a set of size n such that every subset B subseteq A with lvert B rvert =4 has lvert B-B rvert geq 11 . Find the best constant c 0 such that A must always contain a Sidon set of s…
Let G n be the unit distance graph in mathbb R ^n , with two vertices joined by an edge if and only if the distance between them is 1 . Estimate the chromatic number chi(G n) . Does it grow exponentially in n ? …
Let A subseteq mathbb R ^d be a set of n points such that all pairwise distances differ by at least 1 . Is the diameter of A at least (1+o(1))n^2 ?
Let F k(n) be minimal such that for any n points in mathbb R ^2 there exist at most F k(n) many distinct lines passing through at least k of the points, and f k(n) similarly but with lines passing through ex…
Is it true that the number of incongruent sets of n points in mathbb R ^2 which maximise the number of unit distances tends to infinity as n to infty ? Is it always 1 for n 3 ?
Consider the triangular lattice with minimal distance between two points 1 . Denote by f(t) the number of distances from any points leq t . For example f(1)=6 , f( sqrt 3 )=12 , and f(3)=18 . Let $x 1, ldots,x n…
Are there, for all large n , some points x 1, ldots,x n,y 1, ldots,y n in mathbb R ^2 such that the number of distinct distances d(x i,y j) is o left( frac n sqrt log n right)?