A large number of m-coloured complete infinite subgraphs

Abstract

Given an edge colouring of a graph with a set of m colours, we say that the graph is m - coloured if each of the m colours is used. For an m -colouring Δ of N ( 2 ) , the complete graph on N , we denote by F Δ the set all values γ for which there exists an infinite subset X ⊂ N such that X ( 2 ) is γ -coloured. Properties of this set were first studied by Erickson in 1994. Here, we are interested in estimating the minimum size of F Δ over all m -colourings Δ of N ( 2 ) . Indeed, we shall prove the following result. There exists an absolute constant α > 0 such that for any positive integer m ≠ { ( n 2 ) + 1 , ( n 2 ) + 2 : n ≥ 2 } , | F Δ | ≥ ( 1 + α ) 2 m , for any m -colouring Δ of N ( 2 ) . This proves a conjecture of Narayanan. We remark the result is tight up to the value of α .