Highly connected multicoloured subgraphs of multicoloured graphs

Abstract

Suppose the edges of the complete graph on n vertices, E ( K n ) , 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 = 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 ⩾ 2 , proving in particular that when s = 2 and r + 1 is a power of 2 then the answer lies between 4 n / ( r + 1 ) - 17 kr ( r + 2 k + 1 ) and 4 n / ( r + 1 ) + 4 , that if r = 2 s + 1 then the answer lies between ( 1 - 1 / r s ) n - 7 r s k and ( 1 - 1 / r s ) n + 1 , and that phase transitions occur at s = ⌈ r / 2 ⌉ and s = Θ ( r ) . We shall also mention some of the more glaring open problems relating to this question.