SAN FRANCISCO — U.S. District Judge Haywood Gilliam Jr. let lawyers for both Pinterest Inc. and Pintrips Inc. argue largely uninterrupted Monday morning as a bench trial got underway in the trademark spat between the social-media site and the Internet travel startup.

Pinterest, the nation’s third most popular online social-media site behind Facebook and Twitter, sued Pintrips in 2013 claiming the startup could have chosen any name, but intentionally selected one that is deceptively close to Pinterest’s. Pintrips, which has developed a tool to let users track airfares across multiple websites, has filed a counterclaim asking that Pinterest’s federally registered trademarks be canceled as generic.