I see increasing recognition of our fair alma mater as OHIO in a national context. I wear a ball cap with an arched OHIO on it when traveling and increasingly I get folks who recognize it and say, "Go Bobcats," or something along that line. Yes, I do sometimes run into folks who think I'm supporting the junior institution, but that is becoming rarer. A strange exception is right here in the State of Ohio. I see more people confused in Cleveland than I find in say South Carolina, where I'll be vacationing in the middle of the next month.
I've been sporting OHIO logo'd apparel in my travels for decades. That has led to interactions with OHIO students/alumni/friends/strangers from Tokyo's Narita Airport to Valletta (Malta's capita) to Paris to Ribeauville to Florida to Atlanta, etc. Just adds to the fun of traveling.