“Late night hot sweaty dancing” is also on the menu.

Its opening party featured a performance by Patti Smith (a singer who also made an appearance at the Public Theater’s restaurant and music venue, Joe’s Pub, according to the lawsuit).

The most flagrant trademark infringement, the lawsuit said, occurred when the Public hotel hosted an extended run of “Carmen: To Havana and Back,” an adaptation of the Broadway musical “Carmen Jones.” The hotel called it an “exhilarating blend of theater and night life.”

Mr. Schrager said in a statement through his spokeswoman that when his company registered its trademarks for the hotel, the Public Theater did not have any of its own. “We would not have gotten our trademarks if they did,” he said.

He said his company had obtained its trademark on the advice of its lawyers. “I can’t speak to what they did after we got ours or in later years,” Mr. Schrager said. “After being in the business for 40 years with scores of projects having been completed, I think I know a little about registering trademarks to protect our brands and good will.”

Ian Schrager Company registered a trademark for “PUBLIC” in 2012, according to the United States Patent and Trademark Office database, the year after Mr. Schrager opened a hotel with that name in Chicago. The trademark is listed as being used in connection with hotel services.

Mr. Willingham of the Public Theater said he had no problem with Mr. Schrager using the name in association with a hotel. But he said a line was crossed when the name and logo was linked to entertainment, including music and theater.