The de Bruijn-Erd\H{o}s 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 Erd\H{o}s, we consider curves in the unit $n$-cube $[0,1]^n$ of the form \[ A=\{(x,f_1(x),\ldots,f_{n-2}(x),\alpha): x\in [0,1]\}, \] 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 $n$-\emph{de~Bruijn-Erd\H{o}s-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-Erd\H{o}s-set whose $1$-dimensional Hausdorff measure equals $n-1$.