Sets of vectors with many orthogonal pairs (Q1205349)

The authors propose the study of the following problem: What is the maximum number of vectors, \(\alpha^{(d)}(k)\), in \(R^ d\) such that any \(k+1\) contain an orthogonal pair? The example of \(k\) orthogonal basis shows \(\alpha^{(d)}(k)\geq dk\) and Erdős conjectured equality here. The authors show \(\alpha^ 4(5)\geq 24\), refuting Erdős' conjecture. It is shown that \[ \lim_{k\to\infty} \alpha^{(d)}(k)/k=\sup_ k \alpha^{(d)}(k)/k. \] Relationship to the chromatic number of subsets of \(R^ n\) is exhibited.