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

@article{Moshkovitz2012RamseyTI,
  title={Ramsey Theory, integer partitions and a new proof of the Erdős–Szekeres Theorem},
  author={Guy Moshkovitz and Asaf Shapira},
  journal={Advances in Mathematics},
  year={2012},
  volume={262},
  pages={1107-1129}
}
Abstract Let H be a k -uniform hypergraph whose vertices are the integers 1 , … , N . We say that H contains a monotone path of length n if there are x 1 x 2 ⋯ x n + k − 1 so that H contains all n edges of the form { x i , x i + 1 , … , x i + k − 1 } . Let N k ( q , n ) be the smallest integer N so that every q -coloring of the edges of the complete k -uniform hypergraph on N vertices contains a monochromatic monotone path of length n . While the study of N k ( q , n ) for specific values of k… Expand

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