Category Archives: Combinatorics

The Cage Problem

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 , , , , , | Leave a comment

Expander Mixing Lemma in Finite Geometry

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 , , , , | Leave a comment

Incidence Bounds and Interlacing Eigenvalues

The Szemerédi–Trotter theorem is one of the central results in discrete geometry which gives us a (tight) bound on the number of incidences, i.e., the number of point-line pairs with the point lying on the line, between finite sets of points and lines … Continue reading

Posted in Combinatorics, Finite Geometry, Incidence Geometry, Spectral Graph Theory | Tagged , , , , , , | 3 Comments

Applications of Alon-Furedi to finite geometry

In a previous post I discussed how the Alon-Furedi theorem serves as a common generalisation of the results of Schwartz, DeMillo, Lipton and Zippel. Here I will show some nice applications of this theorem to finite geometry (reference: Section 6 of my … Continue reading

Posted in Combinatorics, Finite Geometry, Polynomial Method | Tagged , , | Leave a comment

The Ellenberg-Gijswijt bound on cap sets

Four days back Jordan Ellenberg posted the following on his blog: Briefly:  it seems to me that the idea of the Croot-Lev-Pach paper I posted about yesterday can indeed be used to give a new bound on the size of subsets … Continue reading

Posted in Combinatorics, Finite Geometry, Polynomial Method | Tagged , , , , , | 6 Comments

The Erdős-Ginzburg-Ziv theorem

Let be  a sequence of integers (not necessarily distinct). Then there exists a subsequence of the sum of whose elements is divisible by .  This is one of the first problems I saw when learning the pigeonhole principle. And it’s … Continue reading

Posted in Combinatorics, Polynomial Method | Tagged , , , | 7 Comments

Alon-Furedi, Schwartz-Zippel, DeMillo-Lipton and their common generalization

In the post Balls in Bins I wrote about a combinatorial function which denotes the minimum value of the product among all distributions of balls (so ) in bins with the constraints . It turns out that this combinatorial function is linked … Continue reading

Posted in Combinatorics, Polynomial Method | Tagged , , , | 2 Comments