Colour with no floor

1 · State the answer

Beginner

What does the paper prove?

That at least three numbers in four are always same-colour products That no minimum share of same-colour products applies to every colouring That a colouring can avoid same-colour products altogether

2 · Count up to 100

Colouring: odd numbers red, even numbers blue. The number of same-colour products up to n is (3n + 1)/4 rounded down to a whole number.

Intermediate

How many are there from 1 to 100?

50 100 75

3 · Apply the mirror rule

In the construction, below a cut-off N, a number x and its mirror N/x always have opposite colours. The real cut-offs are gigantic; apply the rule in miniature, with N = 100, and suppose 4 is red.

Expert

What colour is 25?

Red Blue It depends on the colour of 1

4 · Measure the unreachable

To get below 10%, the paper computes that, if 1 is red, the first 10^(10^41,823) numbers must all be red: the first cut-off lies beyond them.

Expert

What follows?

That no computer will ever observe the effect by counting That the proof is wrong, since all those numbers are red That the share of same-colour products stays below 10% from the very first numbers

Floor broken

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