William O. J. Moser

A survey of Sylvester's problem and its generalizations

SummaryLetP be a finite set of three or more noncollinear points in the plane. A line which contains two or more points ofP is called aconnecting line (determined byP), and we call a connecting lineordinary if it contains precisely two points ofP. Almost a century ago, Sylvester posed the disarmingly simple question:Must every set P determine at least one ordinary line? No solution was offered at that time and the problem seemed to have been forgotten. Forty years later it was independently rediscovered by Erdös, and solved by Gallai. In 1943 Erdös proposed the problem in the American Mathematical Monthly, still unaware that it had been asked fifty years earlier, and the following year Gallais solution appeared in print. Since then there has appeared a substantial literature on the problem and its generalizations.In this survey we review, in the first two sections, Sylvesters problem and its generalization to higher dimension. Then we gather results about the connecting lines, that is, the lines containing two or more of the points. Following this we look at the generalization to finite collections of sets of points. Finally, the points will be colored and the search will be for monochromatic connecting lines.

Problems, problems, problems

Collection de problemes (classiques) en geometrie combinatoire, presentee a la Premiere Conference Canadienne en Geometrie algorithmique en 1989

Optimal crossing-free Hamiltonian circuit drawings of Kn☆

The number of crossing-free Hamiltonian circuits in planar drawings of Kn is studied. In particular, it is shown that for planar drawings of Kn, (1) there are drawings having as many as (320)·10[n3] such circuits and (2) no drawing contains more than 2·6n−2([n2])!

Generalizations of Terquem's problem

Abstract Terquems problem asks for the number of combinations 1≤ x 1 2 p ≤n with even entries in even position and odd entries in odd position. This is generalized to x 1 ≡1+ k 1 (mod m ), x j ≡x j-1 +1+ k j (mod m ), j =2(1) p . The case m =2, k j =0, j =1(1) p is Terquems problem, while k j =0, j =1(1) p is Skolems generalization. Also considered is the “circular” type problem which results when the condition x 1 ≡ 1+k 1 (mod m ) is deleted.

Enumeration of combinations with restricted differences and cospan

Abstract For the p -combination 1≤ x 1 x 2 x p ≤ n the differences are d j =x j+1 −x j , j =1(1) p −1, the span is d=x p −x 1 , and the cospan is n−d . An explicit expression (14) is obtained for the number of p -combinations satisfying the conditions k≤d j ≤k′ , j =1(1) p −1, l≤n−d≤l′ . Recurrence relations are found. Special cases include generalized Fibonacci and Lucas numbers.

PROBLEMS ON EXTREMAL PROPERTIES OF A FINITE SET OF POINTS

In 1977 I started a collection of problems in discrete geometry, distributing it and the subsequent revised versions to anyone who requested a copy. The most recent edition, “Research Problems in Discrete Geometry, 1981” (RPDG 1981) contains 68 problems; here I focus on problems from this collection that investigate extremal properties of a finite set of points. A revised version of each problem is given here to include whatever new information has become available. The designation number (# n) locates the problem in RPDG 1981.

On the relative widths of coverings by convex bodies

The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al (5; see also 1; 4; 6; 7; 8).

The number of subsets without a fixed circular distance

An explicit formula is derived for the number of k-element subsets A of {1,2,…, n} such that no two elements in A are at “circular” distance q, i.e., if i ϵ A and 1 ⩽ i ⩽ n − q (resp. n − q + 1 ⩽ i ⩽ n) then i + q ∉ A (resp. i + q − n ∉ A).

A generalization of Riordan's formula for 3x n latin rectangles

A formula is obtained for the number of three-line latin rectangles with first row normalized and second row a permutation which, in disjoint cycle form, has no cycles of prescribed lengths.

Almost all trees have tribe number 2 or 3

Abstract Let T be a tree on n vertices, and let e ⩽ 1 2 be a small fixed positive number. The tribe number tT(ϵ) of T is the smallest integer r such that when any vertex is deleted, some r or fewer subtrees in the resulting forest together contain more than (1−e)n vertices. We prove the following, theorem: Almost all trees have tribe number 2 or 3.