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We study a P\'olya-type urn model defined as follows. Start at time 0 with asingle ball of some colour. Then, at each time n>0, choose a ball from the urnuniformly at random. With probability 1/2<p<1, return the ball to the urn alongwith another ball of the same colour. With probability 1-p, recolour the ballto a new colour and then return it to the urn. This is equivalent to thesupercritical case of a random graph model studied by Backhausz and M\'ori, andTh\"ornblad. We prove that, with probability 1, there is a dominating colour,in the sense that, after some random but finite time, there is a colour thatalways has the most number of balls. A crucial part of the proof is theanalysis of an urn model with two colours, in which the observed ball isreturned to the urn along with another ball of the same colour with probabilityp, and removed with probability 1-p. Our results here generalise a classicalresult about the P\'olya urn model (which corresponds to p=1).

The dominating colour of an infinite P\'olya urn model