In order to establish this partition relation, Todorčević discovered an entirely new mathematical object called rho functions. Sierpinski in 1933 coloured the edges of the complete graph formula_9 whose vertices are the elements of the smallest uncountable cardinal number. He coloured the edges of formula_9 with 2 colours in such a way that each colour appears on some edge of any uncountable subgraph of formula_9. Galvin and Shelah in 1980s had increased the number of colours from 2 to 3. Improving 3 to 4 seemed beyond any available methods. Todorčević used his newly discovered rho functions to increase the colours not just to 4, but all the way up to the smallest uncountable cardinal, which is the maximum conceivable number. This was one of the results for which he was invited to the Berlin ICM.