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Tag Archives: ramsey numbers
A short video on Ramsey numbers
I was recently involved in making a 1 minute maths video for a contest organised by Veritasium. Here is the main requirement for the video We are looking for videos that clearly and creatively explain complex or counterintuitive concepts in … Continue reading
Posted in Combinatorics, Ramsey Theory
Tagged combinatorics, extremal problems, ramsey, ramsey numbers, Veritasium, Youtube
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Bilinear forms and diagonal Ramsey numbers
The recent breakthrough of Conlon and Ferber has shown us that algebraic methods can be used in combination with probabilistic methods to improve bounds on multicolour diagonal Ramsey numbers. This was already shown for the offdiagonal Ramsey numbers by Mubayi … Continue reading
Improved lower bounds for multicolour diagonal Ramsey numbers
Big news in combinatorics today: David Conlon and Asaf Ferber have posted a 4page preprint on arXiv that gives exponential improvements in the lower bounds on multicolour diagonal Ramsey numbers, when the number of colours is at least (also see … Continue reading
Generalized polygons in extremal combinatorics
Jacques Tits invented generalized polygons to give a geometrical interpretation of the exceptional groups of Lie type. The prototype of these incidence geometries already appears in his 1956 paper, while they are axiomatically defined in his influential 1959 paper on … Continue reading
Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Incidence Geometry, Ramsey Theory, Uncategorized
Tagged extremal problems, generalized polygons, generalized quadrangles, Graph Theory, hypergraphs, John Bamberg, ramsey, ramsey numbers, research, Thomas Lesgourgues, turan problems
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Bounds on Ramsey numbers from finite geometry
In an earlier post I talked about the work of Mubayi and Verstraete on determining the offdiagonal Ramsey numbers via certain optimal pseudorandom graphs, which are not yet known to exist except for the case of triangles. Beyond this conditional … Continue reading
Kopparty’s graph
Alon’s construction of optimal pseudorandom graphs from 1994 is useful for obtaining several interesting combinatorial results in various areas of mathematics, some of which are highlighted in this survey of Noga from last year (also see my previous post). In … Continue reading
Ramsey numbers from pseudorandom graphs
One of the foundational results in modern combinatorics is Ramsey’s theorem which states that for every positive integers there exists a constant such that for all , every coloring of edges of the complete graph has a monochromatic copy of … Continue reading