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Amit, Linial, and Matou\vsek (Random lifts of graphs III: independence andchromatic number, Random Struct. Algorithms, 2001) have raised the followingquestion: Is the chromatic number of random $h$-lifts of $K_5$ asymptotically(for $h\to\infty$) almost surely equal to a single number? In this paper, weoffer the following partial result: The chromatic number of a random lift of$K_5\setminus e$ is asymptotically almost surely three.

Random lifts of $K_5\setminus e$ are 3-colourable