Increasing Sequences of Integer Triples

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• Jason Long (University of Cambridge)
• Thursday 27 October 2016, 14:30-15:30
• MR12.

If you have a question about this talk, please contact Andrew Thomason.

We will consider the following deceptively simple question, formulated recently by Po Shen Loh who connected it to an open problem in Ramsey Theory. Define the ‘2-less than’ relation on the set of triples of integers by saying that a triple x is 2-less than a triple y if x is less than y in at least two coordinates. What is the maximal length of a sequence of triples taking values in {1,...,n} which is totally ordered by the ‘2-less than’ relation?

In his paper, Loh uses the triangle removal lemma to improve on the trivial upper bound of n² by a factor of log*(n), and conjectures that the truth should be of order n^(3/2). The gap between these bounds has proved to be surprisingly resistant. We shall discuss joint work with Tim Gowers, giving some developments towards this conjecture and a wide array of natural extensions of the problem. Many of these extensions remain open.

This talk is part of the Combinatorics Seminar series.

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