Problems
Show that \(R(4,4)\ge18\) - that is, there’s a way of colouring the edges of \(K_{17}\) such that there’s no monochromatic \(K_4\).
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 \(R(k,k)\le 4^k\).
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?
The sum of Matt’s and Parker’s ages is \(63\) years. Matt is twice as old as Parker was when Matt was as old as Parker is now. How old are they? (Show that there’s no other ages that they could have)