Ramsey Theory, integer partitions and a new proof of the Erdős–Szekeres Theorem

Let H be a k  -uniform hypergraph whose vertices are the integers 1,…,N1,…,N. We say that H contains a monotone path of length n   if there are x1• We show that for every fixed q   we have N3(q,n)=2Θ(nq−1)N3(q,n)=2Θ(nq−1), thus resolving an open problem raised by Fox et al.
• We show that for every k≥3k≥3, Nk(2,n)=2⋅⋅2(2−o(1))nNk(2,n)=2⋅⋅2(2−o(1))n where the height of the tower is k−2k−2, thus resolving an open problem raised by Eliáš and Matoušek.
• We give a new pigeonhole proof of the Erdős–Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdős–Szekeres Lemma on increasing/decreasing subsequences.