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Article
Mathematics
A strengthening of the Erdos-Szekeres Theorem
Jozsef Balogh et al.
Summary: The Erdos-Szekeres Theorem in terms of graphs states that any red-blue coloring of the edges of the ordered complete graph Krs+1 must contain either a red monotone increasing path with r edges or a blue monotone increasing path with s edges. Subgraphs with this coloring property are characterized by containing all the edges of a graph called the circus tent graph CT(r, s). Additionally, similar proof techniques are used to enhance the bounds on the online ordered size Ramsey number of a path.
EUROPEAN JOURNAL OF COMBINATORICS (2022)
Article
Mathematics
On-line size Ramsey number for monotone k-uniform ordered paths with uniform looseness
Xavier Perez-Gimenez et al.
Summary: An ordered hypergraph is a hypergraph with a specified linear ordering of vertices, and the appearance of G in H must respect the specified order. In online Ramsey theory, Builder presents edges for Painter to immediately color, aiming to force a monochromatic copy of G using t colors. Our bounds on (R) over tilde (t)(P-r((k))) improve prior results for t growing faster than m/logm.
EUROPEAN JOURNAL OF COMBINATORICS (2021)
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