We see a number of different relationships in Chapter Three of Theodore Taylor's The Cay. A relationship is simply any connection or association between two people. Hence, any time we see two characters interact, we see a relationship between those characters.
We see one relationship with respect to Phillip and the captain of the ship. Phillip notes that the passengers had been put through many drills in order to know what to do in the event of an emergency. Hence, Phillip is not surprised by any of the activity as the crew prepares to get passengers in the lifeboat. He particularly notes that the captain "was acting the way [he'd] been told captains should act" (p. 29). He continues to describe the captain as standing by the lifeboat, lit by the "fire's glow" as it burns on-board the ship, and giving orders to the crew. Since Phillip depends on the captain for survival as a passenger on the ship, we can see that Phillip has a relationship with the captain. Plus, the above passage shows us how the captain is upholding that relationship by giving orders necessary for survival.
A second relationship in this chapter is between Phillip and the black man, named Timothy, who rescued him. At the start of the novel, Phillip is filled with the prejudices he has been taught by his parents; therefore, he finds Timothy to be ugly and hates the fact that he is on a raft alone with him instead of with his mother. Hence, at the beginning of the novel, the relationship between Phillip and Timothy is a very distant relationship, despite the fact that Phillip owes Timothy his life. The tension in their early relationship and the unhappiness Phillip feels is expressed in the following passage:
That lonely sea, and the sharp pains in my head, and the knowledge that I was here alone with a black man instead of my mother made me break into tears. (p. 32)
Since Phillip does not want to be alone with Timothy and wants his mother with him instead, we can see how the above passage reflects Phillip's current relationship with Timothy.