Foreteen is the term used to describe the relationship between Eric Foreman and Remy "Thirteen" Hadley. Unlike the other terms fans use to describe relationships, this term was actually used on the series. The relationship lasted from Season 5 until Season 6, but it's ramifications extended well into Season 7.
The genesis of the relationship was in the episode Last Resort. The patient takes Thirteen and a group of other patients hostage and uses Thirteen as a guinea pig to ensure none of the medication the doctors are using is a sedative, and she suffers a number of adverse side effects from the drugs which damage her kidneys and require convalescence. Foreman, who was not directly involved in the hostage taking, is seen by her bed while she recovers. When she wakes and sees Foreman there, she finally agrees to enter the Huntington's disease clinical trial Foreman is urging on her.
As Foreman and Thirteen see each other frequently during the clinical trial. However, when Thirteen starts being late for her appointments, Foreman breaks in to her apartment to determine if her self-destructive behavior has started again. Instead, he finds that Thirteen has been following the instructions he gave her to the letter. When she returns, he confronts her and she admits she's late for appointments to avoid the other patients whose symptoms remind her of what's in store for her. Foreman tells her to be on time from now on and to stop feeling sorry for herself.
Thirteen still has problems confronting her fears, but finally befriends Janice Burke, an older patient who has the appointment before her. She finally admits to Foreman that she was not avoiding Janice because of her fear of the future, but because Janice reminded her of her own mother and when her mother was taken for institutional care and she refused to go along with her. The incident brings the two closer.
At Christmas, Thirteen gets Foreman a gift - a day at the spa. The two start to joke to pass the time during the appointments for the clinical trial. However, they get into a fight about Janice - Foreman was rude to her when Janice complained about nausea and Janice dropped out of the clinical trial. At first, Foreman defends his actions, but after realizing Thirteen was right about him acting like House, he apologizes to Janice and gets her into another trial. That evening, instead of going to the hospital's Christmas party, Foreman and Thirteen stay behind and make out.
However, Thirteen immediately pulls back. Foreman makes it clear he knows that Thirteen's health will be a challenge in the future, but Thirteen says she doesn't want to pull him down with her. It doesn't help when she learns Janice is improving because Janice's appointment was moved to be immediately before Thirteen's. She confronts Foreman, who denies knowing about the switch in appointments, but in fact Foreman did arrange it. Thirteen never learns and decides to ask Foreman out on a date.
Their relationship blossoms, but problems develop when Foreman accidentally learns Thirteen is on the placebo. After asking for advice, he gets a solid "don't be stupid", but House realizes he's in love with Thirteen and is going to do it anyway. Foreman switches Thirteen's medication so she will be receiving the actual medicine. However, Thirteen soon develops severe headaches that develop into reduced peripheral vision and then blindness. Foreman has to tell her about his deception in order to get her to agree to an examination. They find a brain tumor, which luckily is treatable. Thirteen is angry with Foreman for making the decision to go ahead with the change in treatment without consulting her. Foreman manages to barely hang on to his medical license by disclosing his deception and giving up on the clinical trial. Foreman and Thirteen quickly reconcile.
However, House figures that their ongoing relationship is interfering with their work and orders them to split or quit. He fires Foreman when they don't comply. However, he gets his job back and they deceive House into thinking they have split up when they are really still together.
When they return to work, Kutner soon deduces their break-up was a sham and informs Taub. They take the opportunity to tease Foreman about Thirteen's promiscuity and tease Thirteen about Foreman's lack of emotion. However, when Kutner comes clean, they can't figure out why House didn't realize they were still together. It turns out House is on methadone, which has dulled his skills. When House walks off the case, Thirteen and Foreman find a vital clue while they are getting ready for work in the morning and, when they return and find House back on the case, it's clear House doesn't care about their relationship anymore.
The relationship is strained somewhat after Kutner's suicide when Foreman withdraws from the world. However, at Kutner's memorial service, Foreman reaches out to Thirteen again.
Going into Season 6, the relationship seems strong, and when Foreman takes over the diagnostic department and Taub quits, leaving the two alone, she encourages him and tells him that finding another fellow will be no problem at all. However, by the end of the case, Foreman feels that Thirteen isn't standing up to him when he's wrong and it's because of their relationship. To save his relationship, he fires her from the team.
Although Foreman is supportive and tries to find Thirteen another job and keep the relationship going, it's clear that the firing has permanently damaged the relationship. Thirteen flies off to Thailand.
However, when House gets back in charge, he manipulates Thirteen into re-joining the team. Things are tense between Foreman and Thirteen, even though they tell everyone who will listen that they no longer have feelings for each other. However, after a particularly bad fight, Foreman apologizes to Thirteen for firing her, admits that he knew it was the wrong decision the minute he made it, and further admits that he was too proud to back off.
Although the relationship is unlikely to resume, it's still clear that Foreman has feelings for Thirteen. He remains very protective of her and is the most upset when she's evasive and secretive with the rest of the team.