The de Bruijn–Erdős theorem from a Hausdorff measure point of view

Abstract

Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erdős, we consider curves in the unit n-cube $[0,1]^n$ of the form

$$A= \bigl\{(x,f_1(x),\ldots,f_{n-2}(x),\alpha): x\in [0,1] \bigr\},$$

where $\alpha$ is a fixed real number in [0,1] and $f_1, \ldots, f_{n-2}$ are injective measurable functions from [0,1] to [0,1]. We refer to such a curve A as an n-de Bruijn–Erdős-set. Under the additional assumption that all functions $f_i, i=1,\ldots,n-2,$ are piecewise monotone, we show that the Hausdorff dimension of A is at most 1 as well as that its 1-dimensional Hausdorff measure is at most n-1. Moreover, via a walk along devil’s staircases, we construct a piecewise monotone n-de Bruijn–Erdős-set whose 1-dimensional Hausdorff measure equals n-1.