Problems

Show that \(R(4,3)\le9\). That is, no matter how you colour the edge of \(K_9\), there must be a red \(K_4\) or a blue \(K_3\).

Show that \(R(4,4)\le18\) - that is, no matter how you colour the edges of \(K_{18}\), there must be a monochromatic \(K_4\).

By considering \(k-1\) copies of \(K_{k-1}\), show that \(R(k,k)\ge(k-1)^2\).

Let \(s>2\) and \(t>2\) be integers. Show that \(R(s,t)\le R(s-1,t)+R(s,t-1)\).

Using \(R(s,t)\le R(s-1,t)+R(s,t-1)\), prove that \(R(k,k)\le 4^k\).

Show that \(R(k,k)\ge2^{k/2}\).

Explain why you can’t rotate the sides on a normal Rubik’s cube to get to the following picture (with no removing stickers, painting, or other cheating allowed).

Are there any two-digit numbers which are the product of their digits?

How many \(10\)-digit numbers are there such that the sum of their digits is \(3\)?

The sum of digits of a positive integer \(n\) is the same as the number of digits of \(n\). What are the possible products of the digits of \(n\)?