By chapter 10, three people have died, and there is no longer any question that they have all been murdered. The remaining guests realize they they, too, are likely to be murdered.
Vera Claythorne and Philip Lombard pair up defensively because neither one thinks the other is the murderer. As Vera states, she can't imagine Philip as the man who "dictated that gramophone record." He agrees, saying he would only kill if he could get something out it. He, in turn, believes the murderer must be insane. Since Vera comes across to him as very sane and sensible, he can't imagine she is killing people. Both of them feel that by teaming up, they will have a better chance of figuring out who the killer is and defending themselves against that person.
Ex-inspector Blore and butler Rogers also pair up out of fear of being murdered. Like Vera and Philip, they would prefer to dismiss the whole problem of a killer on the loose as a bad dream, but they know they can't. Both are very nervous.
Dr. Armstrong and Wargrave also team up. Dr. Armstrong assesses Judge Wargrave as a man, who although old, is very concerned to preserve his life, and therefore a good ally. They are also of a similar social class, which makes them likely partners: both are educated, professional men.
Only Miss Emily Brent is isolated and alone.
All of the characters who team up do so because they want to save themselves. They are also frightened and take comfort in having someone else to talk to about the murders. Having an ally makes them feel less isolated and alone. It is telling that the pairings make sense: Vera and Philip feel a physical attraction to each other, Armstrong and Wargrave are of a similar social class, and Blore and Rogers, while not of the same social class (Roger's Cockney language shows he is of a lower class than Blore) are neither very high up the class ladder. They each pick a partner they can feel comfortable with. Miss Brent's isolation is indicative of her cold character.