What is a red airway on the VFR sectional?
These are Red Federal Airways, and can be found in Order 7400.9

Red airways are one of a 1950s-ish system of airways: Red, Blue, Green, Amber, which are designated by an NDB (rather than a VOR or VORTAC for a V or J route).

Green and Red airways go east/west, and Blue and Amber routes go north/south. 

These routes were prioritized: GARB, with Green routes having the highest priority.