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Jason Long (University of Cambridge)
Thursday 27 October 2016, 14:30-15:30
Increasing Sequences of Integer Triples
- đ¤ Speaker: Jason Long (University of Cambridge)
- đ Date & Time: Thursday 27 October 2016, 14:30 - 15:30
- đ Venue: MR12
Abstract
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.
Series This talk is part of the Combinatorics Seminar series.
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