Anurag's Math Blog

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Mathematics is political
Sep 11, 2026 · original
I have often heard mathematicians saying, “We do not like politics” and “We just want to focus on research”. This is typically a reaction to forces or actions that could potentially challenge the privileges that they have enjoyed so far, or, if I am being charitable, a genuine misunderstanding of what “political” means. The recent involvement of Big Tech in mathematical research is igniting both soul-searching in our community and direct political movements. For example, the Leiden declaration , The Association for Human Mathematics (AHM) , \begin{proof} , and Math and AI . This is a welcome change, and I hope that the community is able to take some immediate actions to counter the influence of Big Tech, and “Big Mathematicians” who endorse it. Here are some articles that I recommend reading: How we can win by Tasmin Chu Knowledge collapse by Michael Harris Mathematics is not rational, r
Postdoc position in finite geometry and Ramsey theory
Jun 27, 2026 · original
We are advertising a 3 year postdoctoral position at the University of Western Australia to work at the interface of finite geometry and Ramsey theory. The project is led by John Bamberg at UWA, Sam Mattheus at Vrije Universiteit Brussel, and me. It includes funds for extended research visits to both TU Delft and VUB. More details, including the application link, can be found here: https://external.jobs.uwa.edu.au/en/job/523511/postdoctoral-research-associate-pure-mathematics Closing date: 11:55 PM AWST (17:55 CEST) on Wednesday, 22 July 2026
Tight bounds on off-diagonal Ramsey numbers
Jun 18, 2026 · original
One of the oldest open problem in Ramsey theory has been solved. We now know the asymptotics of Ramsey numbers , for any fixed , up-to log factors. Last month, Domagoj Bradač showed that for any fixed and (where the term is hiding log factors), using a beautiful finite-geometric construction. This is very close to the classical upper bound, which is (roughly) and it drastically improves the previous best lower bound of for all . Now an internal model at openai has been used by the mathematicians working at openai to close the gap, as described in the new version of Domagoj’s paper. The construction is slightly different from the original one by Domagoj. Perhaps expectedly, it uses an old explicit construction of Codenotti, Pudlák and Resta for Ramsey numbers as the base geometric graph and then does a random sampling via a similar, but more involved, argument. In this post I’ll leave out
Finite geometry paves the way for another Ramsey breakthrough
May 30, 2026 · original
A sensational Ramsey breakthrough by Domagoj Bradač
Small complete cap sets
Mar 10, 2026 · original
The following post is (partly) based on discussions with Dion Gijswijt and Ananth Ravi from September 2025. The famous cap set problem has attracted the attention of various mathematicians from different areas over the last few decades. It asks for the largest possible size, , of a 3-term arithmetic progression free subset of the abelian group , or geometrically the largest size of a points set in the affine geometry where no three points are collinear. In 1982, Brown and Buhler gave the first non-trivial upper bound of . In 1995, Meshulam improved their upper bound to by using Fourier analytic techniques. Bateman and Katz further refined these techniques and managed to prove the upper bound of . Then came the big breakthrough of Ellenberg and Gijswijt in 2016, who used polynomial rank arguments (building up on the work of Croot, Lev and Pach ), to prove that . These results initiated th
Large induced matchings and minimal blocking sets using parabolas
Mar 2, 2026 · original
In this post, I will discuss a recent breakthrough of Hunter, Pohoata, Verstraete and Zhang on an old problem in finite geometry, which also has interesting consequences outside this area. For example, it improves constructions in a minimal distance problem in and it has been adapted by Ihringer and Zhou to give an improved construction of certain locally repairable codes . The problem is to find large induced matchings in the bipartite incidence graph of finite projective planes . This has been studied at least since 1991, when Illés, Szőnyi and Wettl investigated these objects under the name of maximal strong representative systems , motivated by the work of Bruen and Thas from 1977 on minimal blocking sets . The point-set corresponding to the induced matching, that is, the vertices of the matching that lie on the side of points, is also called a tangency set . It has the property that
PhD and postdoc positions in Finite Geometry
Dec 2, 2025 · original
I am hiring people on my NWO Vidi project . There is one PhD position (4 years) and one postdoc position (2 years) that you can apply for if you are interested. Use the following links to get more information and apply: PhD: https://careers.tudelft.nl/job/Delft-PhD-Position-Discrete-Mathematics-Extremal-Problems-in-Finite-Geometry-2628-CD/1333394457/ Postdoc: https://careers.tudelft.nl/job/Delft-Postdoc-in-Discrete-Mathematics-Extremal-Problems-in-Finite-Geometry-2628-CD/1333394757/ Deadline : 1 February 2026. Here are some of my recent papers that are representative of the project: The chromatic number of finite projective spaces, with Wouter Cames van Batenburg and Ananthakrishnan Ravi. arXiv . Ramsey numbers and extremal structures in polar spaces, with John Bamberg, Ferdinand Ihringer and Ananthakrishnan Ravi. arXiv . Explicit constructions of optimal blocking sets and minimal codes,
Coloring projective spaces and Ramsey theory
Dec 2, 2025 · original
What is the minimum number of colors needed to color the points of the Fano plane such that there is no monochromatic line? It is a nice exercise to prove that two colors do not suffice. This fact has been known at least since the 1956 paper of Richardson that introduced the notion of blocking sets in projective spaces, and it commonly appears in the extremal combinatorics literature in the context of Property-B and minimal non-2-colorable hypergraphs (see for example, Section 1.3 in the notes on Probabilistic Combinatorics by Yufei Zhao ). Here is a proper three-coloring of the points, which then shows that the chromatic number of is equal to . . Interestingly, for every larger projective plane two colors do suffice. One way to see this is to take three non-concurrent lines (a triangle), color all points on these lines except for the intersection points (vertices of the triangle) red, a
Circular Sorting
Nov 4, 2025 · original
How many swaps do you need to sort objects on a circle in clockwise order? This fairly simple and natural question quickly leads to some deep mathematics that I would like share. Let’s start with an example for : After swapping 2 with 6, 1 with 4, and then 4 with 5, we get the following `sorted’ arrangement: It turns out that for , every arrangement can be sorted in at most three moves. The natural extremal question is then to study max number of swaps required to sort any circular permutation of . This problem was very recently introduced by Adin, Alon and Roichman who proved various interesting bounds on . In particular, they showed that for all , with equality for prime. They also made the following two conjectures. Conjecture 1 : if and only if is a prime number. Conjecture 2 : For primes , the only circular permutations that required swaps to sort are the affine permutations. An aff
Constructing blocking sets using expander graphs and hypergraphs
Nov 18, 2024 · original
Blocking sets are one of the central topics in finite geometry, which was originally introduced in the context of game theory under the name of `blocking coalitions’: On Finite Projective Games . I first learned about them during my Ph.D. as a source of extremal problems that over the years as led to the development of techniques like the polynomial method and the variance trick . The latter can sometimes be replaced by spectral arguments like the expander mixing lemma, as shown here . All of these interesting methods are only used for proving lower bounds on their size (or upper bounds in case of minimal blocking sets), the constructions have always been geometrical. That changed recently, when Alessandro Neri proposed a graph-theoretical approach to constructing something known as a strong blocking set in his talk at the 2022 Combinatorics conference in Mantua. Using his initial idea,

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