Flight attendants at Delta Air Lines Inc. overwhelmingly rejected union representation, turning back one of labor’s biggest organizing efforts in more than 30 years.

About 5,520 voted for and 89 against joining the Association of Flight Attendants, officials announced late Friday. But federal labor law requires that the union gain endorsement from a majority of Delta’s 19,033 covered employees to gain bargaining recognition. Under the law, a ballot that isn’t cast is counted as a no vote.

Delta, the nation’s third-largest airline, is one of the least unionized in the industry. It is the only major airline whose flight attendants are not unionized. The pilots are Delta’s only major employee group represented by a union.

A victory by the AFA would have provided a huge boost for the entire labor movement, which today represents less than 10 percent of the nation’s private work force.

The AFA said it would immediately challenge the results on grounds that the company illegally interfered with the election. The union said supervisors illegally questioned flight attendants about their support for the union, sent letters and videos to their homes warning that jobs were at stake, and held one-on-one meetings with them in which they were accused of being “anti-Delta.”

AFA President Patricia Friend said, “Delta’s entire campaign focused on creating fear and uncertainty in flight attendants’ minds.”

The National Mediation Board, which supervised the election, said it would investigate the allegations. If the board finds the company illegally interfered with the election, it can set aside the result and order a new vote.

Delta Chairman Leo Mullin, who last week said he was confident the company’s victory would be overwhelming, said he was “enormously gratified” that the flight attendants had decided to remain “union free.”

Because of the way airline union votes are counted under federal law, Delta developed what has been described as a “rip it up” campaign in which they encouraged flight attendants to simply tear up their ballots and not cast a vote.