Tag Archives: finite geometry

A coding theoretic application of the Alon-Füredi theorem

Posted on March 7, 2021 by Anurag Bishnoi

The Alon-Füredi theorem is something that I have written a lot about in this blog. I spent a considerable amount of time on this theorem during my PhD. In fact, it’s generalisation that I obtained and it’s applications in finite … Continue reading →

Posted in Coding Theory, Combinatorics, Extremal Combinatorics, Finite Geometry, Polynomial Method | Tagged Alon-Furedi, finite fields, finite geometry, linear codes, minimal codes, polynomials | 2 Comments

Bilinear forms and diagonal Ramsey numbers

Posted on November 3, 2020 by Anurag Bishnoi

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 off-diagonal Ramsey numbers by Mubayi … Continue reading →

Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Ramsey Theory | Tagged Asaf Ferber, bilinear forms, David Conlon, finite fields, finite geometry, othogonality, polar spaces, ramsey numbers | Leave a comment

Improved lower bounds for multicolour diagonal Ramsey numbers

Posted on September 23, 2020 by Anurag Bishnoi

Big news in combinatorics today: David Conlon and Asaf Ferber have posted a 4-page 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 →

Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Incidence Geometry, Ramsey Theory | Tagged Asaf Ferber, David Conlon, finite fields, finite geometry, polar spaces, ramsey, ramsey numbers | 20 Comments

Heisenberg groups, irreducible cubics and minimal Ramsey

Posted on September 10, 2020 by Anurag Bishnoi

As I mentioned in a previous post, we recently improved the upper bound on a Ramsey parameter, in collaboration with John Bamberg and Thomas Lesgourgues. My favourite thing about this work is how it ends up using the properties of … Continue reading →

Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Incidence Geometry, Ramsey Theory | Tagged Bill Kantor, finite geometry, generalized quadrangles, Heisenberg groups, irreducible cubics, John Bamberg, minimal Ramsey, ramsey, Thomas Lesgourgues | 2 Comments

Bounds on Ramsey numbers from finite geometry

Posted on January 11, 2020 by Anurag Bishnoi

In an earlier post I talked about the work of Mubayi and Verstraete on determining the off-diagonal 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 →

Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Incidence Geometry, Ramsey Theory | Tagged combinatorics, Dhruv Mubayi, extremal problems, finite geometry, generalized polygons, generalized quadrangles, Graph Theory, Jacques Verstraete, ramsey numbers | 3 Comments

Ryser’s conjecture

Posted on September 20, 2018 by Anurag Bishnoi

I am on a research visit in Rome, working with Valentina Pepe, and our joint paper on Ryser’s conjecture is on arXiv now. So this seems like the right time to talk about the conjecture and the problems related to … Continue reading →

Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Incidence Geometry | Tagged blocking set, combinatorics, finite geometry, Graph Theory, hypergraphs, research, Ryser, Valentina Pepe, vertex cover | 2 Comments

Wenger graphs

Posted on April 28, 2018 by Anurag Bishnoi

A central (and foundational) question in extremal graph theory is the forbidden subgraph problem of Turán, which asks for the largest number of edges in an -vertex graph that does not contain any copy of a given graph as its … Continue reading →

Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Incidence Geometry | Tagged combinatorics, extremal problems, finite geometry, forbidden subgraphs, generalized polygons, generalized quadrangles, Graph Theory, Wenger graphs | 2 Comments

Introduction to polynomial method

Posted on November 25, 2017 by Anurag Bishnoi

(The following is a blog-friendly version of Chapter 7 of my PhD thesis, which is an introduction to the so-called polynomial method.) The polynomial method is an umbrella term for different techniques involving polynomials which have been used to solve … Continue reading →

Posted in Combinatorics, Incidence Geometry, Polynomial Method | Tagged additive-combinatorics, combinatorial nullstellensatz, combinatorics, finite geometry, polynomials | 2 Comments

What I have learned in finite geometry

Posted on August 13, 2017 by Anurag Bishnoi

On September 2nd, 2014 I wrote a blog post titled learning finite geometry, in which I described how much I have learned in my first year of PhD and more importantly, the topics that I wish to learn while I … Continue reading →

Posted in Combinatorics, Finite Geometry, Incidence Geometry, Research Diary | Tagged blocking set, combinatorics, finite geometry, learning, PhD, polar spaces, unitals | Leave a comment

The cage problem and generalized polygons (part 1)

Posted on August 9, 2017 by Anurag Bishnoi

This post is a continuation of my previous post on the cage problem. Just to recall the main problem, for any given integers and , we want to find the least number of vertices in a simple undirected graph which … Continue reading →

Posted in Combinatorics, Extremal Combinatorics, Finite Geometry, Incidence Geometry, Spectral Graph Theory | Tagged cage, expander mixing lemma, finite geometry, Graph Theory, Moore graphs | Leave a comment