Published online by Cambridge University Press: 20 November 2018
In a finite graph with no loops nor multiple edges, two points a and b are said to be connected by an r-way, or more explicitly, by a line r-way a — b if there are r paths, no two of which have lines in common (although they may share common points), which join a to b. In this note we demonstrate that any graph with n points and 3n — 2 or more lines must contain a pair of points joined by a 6-way, and that 3n — 2 is the minimum number of lines which guarantees the presence of a 6-way in a graph of n points.
In the language of [3], this minimum number of lines needed to guarantee a 6-way is denoted U(n). For the background of this problem, the reader is referred to [3].