A combinatorial problem on polynomials and rational functions
Let \(C\) be a real number with \(C>1\). Let \(s\), \(t\) be two straight lines in the Euclidean plane, while \(U\), \(V\) are two collinear sets of \(n\) points each, located on \(s\), \(t\) respectively. Further, assume that among the \(n^2\) distances \(d(U_i,V_j)\) for \(U_i\in U\) and \(V_j\in V\), only some \(C_n\) are distinct. G. Purdy conjectured that if \(n> n_0(C)\), then the two lines must be parallel or orthogonal. This conjecture is related to a famous problem of \textit{P. Erdős} [Am. Math. Mon. 53, 248-250 (1946; Zbl 0060.34805)]. Using the characterization of restricted functions, the present authors settle this conjecture in the affirmative.
scientific article; zbMATH DE number 1421328