Tag Archives: finite geometry

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 | 1 Comment

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 | 1 Comment

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

The Cage Problem

Posted on April 14, 2017 by Anurag Bishnoi

I recently finished my research visit to UWA where I worked with John Bamberg and Gordon Royle on some finite geometrical problems related to cages. So this seems like the right time for me to write a blog post about … Continue reading →

Posted in Combinatorics, Finite Geometry | Tagged cage, finite geometry, Gordon Royle, Graph Theory, John Bamberg, Moore graphs | 2 Comments

Expander Mixing Lemma in Finite Geometry

Posted on April 2, 2017 by Anurag Bishnoi

In this post I will discuss some nice applications of the expander mixing lemma in finite incidence geometry, including a new result that I have obtained recently. In many of the applications of the lemma in finite geometry, the graph is bipartite, and … Continue reading →

Posted in Combinatorics, Finite Geometry, Incidence Geometry, Spectral Graph Theory | Tagged blocking set, combinatorics, expander mixing lemma, finite geometry, Graph Theory | 3 Comments

Generalized hexagons containing a subhexagon

Posted on July 5, 2016 by Anurag Bishnoi

I have recently uploaded a joint paper with Bart, “On generalized hexagons of order and containing a subhexagon”,on arXiv and submitted it for publication. In this work we extend the results of my first paper, which I discussed here, by proving the following: … Continue reading →

Posted in Incidence Geometry | Tagged finite geometry, generalized polygons, geometric hyperplanes, semi-finite hexagons | Leave a comment

A timeline of the polynomial method up-to combinatorial nullstellensatz

Posted on August 27, 2015 by Anurag Bishnoi

Over the past 30-40 years, the so-called polynomial method has developed into a powerful tool in combinatorics and (additive) number theory. There has been a lot of recent interest in it after Dvir’s paper on the Kakeya conjecture, where he … Continue reading →

Posted in Combinatorics, Finite Geometry, Polynomial Method | Tagged combinatorics, finite fields, finite geometry, polynomials | 1 Comment