Highly connected multicoloured subgraphs of multicoloured graphs

Suppose the edges of the complete graph on n   vertices, E(Kn)E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s   of the colours? This question is due to Bollobás, and the case s=1s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s⩾2s⩾2, proving in particular that when s=2s=2 and r+1r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1)4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+44n/(r+1)+4, that if r=2s+1r=2s+1 then the answer lies between (1-1/rs)n-7rsk and (1-1/rs)n+1, and that phase transitions occur at s=⌈r/2⌉s=⌈r/2⌉ and s=Θ(r). We shall also mention some of the more glaring open problems relating to this question.