Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree 7t + 7 contains K t as an immersion and that every graph with chromatic number at least 3.54t + 4 contains K t as an immersion. We also show that every graph on n vertices with no stable set of size three contains K 2⌊n/5 ⌋ as an immersion.Hence (i) holds for X 3 .By the same arguments, (i) and (ii) hold for X 1 . This proves the claim. ♦ Claims 5.3 and 5.5 complete the proof of Theorem 1.6.