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A combinatorial problem in geometry
Our present problem has been suggested by Miss Esther Klein in connection with the following proposition.
2. Notation The letters a, b, c, d, x, y, z denote finite sets of non-negative integers, all other lower-case letters denote non-negative integers. If fc I, then [k, I) denotes the set
Some remarks on the theory of graphs
The present note consists of some remarks on graphs. A graph G is a set of points some of which are connected by edges. We assume here that no two points are connected by more than one edge. The
On a lemma of Littlewood and Offord
Remark. Choose Xi = l, n even. Then the interval ( — 1, + 1 ) contains Cn,m s u m s ^ i e ^ , which shows that our theorem is best possible. We clearly can assume that all the Xi are not less than 1.
On Sets of Distances of n Points
1. The function f(n). Let [P. ] be the class of all planar subsets P. of n points and denote by f(n) the minimum number of different distances determined by its n points for P,, an element of { P. }.
On extremal problems of graphs and generalized graphs
  • P. Erdös
  • Mathematics, Computer Science
  • 1 September 1964
It is proved that to everyl andr there is anε(l, r) so that forn>n0 everyr-graph ofn vertices andnr−ε( l, r), which means that all ther-tuples occur in ther-graph.
Graph Theory and Probability
A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete
Some Problems on Random Walk in Space
Consider the lattice formed by all points whose coordinates are integers in d-dimensional Euclidean space, and let a point S j(n) perform a move randomly on this lattice according to the following