Measures of graphs on the reals

Abstract:

This paper studies measure properties of graphs with infinitely many vertices. Let $[0,1]$ denote the real unit interval, and $F$ be the collection of bijections taking $[0,1]$ onto itself. Given a graph $G = \left \langle {[0,1],E} \right \rangle$ and $f \in F$, define the $f$-representation of $G$ to be the set ${E_f} = \{ \langle {f(x),f(y)} \rangle :x,y \in [0,1]$ and $\langle {x,y} \rangle \in E\}$. Let $\mu$ be $2$-dimensional Lebesgue measure. Define the measure spectrum of $G$ to be the set $M(G) = \{ m \in [0,1]:\exists f \in F$ with ${E_f}$ measurable and $\mu {E_f} = m\}$. Our main result characterizes those subsets of reals that are the measure spectra of graphs.