That leaves the union that represents baggage handlers and mechanics with time on its side.
But in one department, car equipment, which represents mechanics and shop workers in the subway system, the vote was extremely close.
Six groups of people appear around the circle, representing war, science, marine, commerce, mechanics, and agriculture.
The airline has not been able to reach agreements with unions representing its flight attendants, mechanics, ground workers and other employees.
The union represents 26,000 baggage handlers, mechanics and other ground personnel at the airline, with which it has been holding talks in recent weeks.
Workers own about 55 percent of the company's stock, with the unions representing United's pilots and mechanics holding the lion's share.
The union representing 1,300 Queens bus drivers and mechanics agreed to a contract yesterday with the city.
Talks continue with flight attendants as well as the union representing mechanics, baggage handlers and other ground workers.
But that same union represents mechanics who work for airlines that work around the clock.
Talks with the Transport Workers Union, which represents mechanics and ground service workers, have continued.