Sequence Erdos-Szekeres: who can see longer mon in
5, 4, 11, 2, 9, 13, 7, 12, 17, 1, 6, 10, 16, 15, 3, 8, 14Â
Thm: If length >=kl+1, then inc of length k+1 or dec of length l+1.
Proof: Write under every number the longest inc and dec sequence starting from it - they are all different.
For a lot more see theory of Young Tableaux.
Ramsey theory: order in chaos If there are six people, prove 3 know each other or 3 don't - why not true for five? - R(3,3)=6
Def: R(k,l) - smallest n such that if we color complete n, there is k clique or l independent from one of the colors
Thm: R(k,l) exists and <={k+l-2\choose k-1} Proof: R(k,l)<= 1+ R(k-1,l)-1+R(k,l-1)-1 +1= R(k-1,l)+R(k,l-1)
Similarly def for more colors.
How big are numbers? R_2(2,k)=k, R_2(k,k)=o(4^k) and \Omega(2^{k/2}).
Lower bound proof by "probabilistic method".