Research

Interests: Finite Geometry, Extremal Combinatorics, Coding Theory, Spectral Graph Theory and Polynomial Methods.

All my preprints are available on arXiv. Also see my google scholar page.

Publications/Preprints

[28] Hat guessing with proper colorings, with Sam Adriaensen, Peter Bentley, Michael Kreiger, Lars van der Kuil, Saptarshi Mandal, Anurag Ramachandran and James Tuite. arXiv.

This project was part of the Polymath Jr program in the summer of 2025. We study the variant of the classic hat guessing game on graphs where the adversary can only choose proper colorings of the graph. We prove several bounds and tight results on this variant. We also relate it to the classical hat guessing number, and propose some interesting open problems. 

[27] The chromatic number of finite projective spaces, with Wouter Cames van Batenburg and Ananthakrishnan Ravi. arXiv. blog.

We make connections between the chromatic number of finite projective spaces (when viewed as a hypergraph) and some problems from Ramsey theory. We prove new bounds on the chromatic numbers, which opens up the doors for making progress on classical multicolor Ramsey number of triangles and proves new bounds on some multicolor vector space Ramsey numbers. 

[26] Circular sorting, strong complete mappings and wreath product constructions, with Paul Bastide, Carla Groenland, Dion Gijswijt, and Rohinee Joshi. arXiv.

We make some progress on the recently introduced problem of sorting points on a circle using transpositions. We provide further evidence for a conjecture of Adin, Alon, and Roichman that for non-prime n the worst case requires at most n – 3 transpositions and disprove their another conjecture that for prime n the worst case permutation need to have an affine structure. To prove these results we establish a relation with strong complete mappings, which are special bijections of finite groups. We also prove some new lower bounds and tight results using a wreath product construction. 

[25] The generalized trifference problem, with Bartłomiej Kielak, Benedek Kovács, Zoltán Lóránt Nagy, Gábor Somlai, Máté Vizer, and Zeyu Zheng, in IEEE Transactions on Information Theory, doi: 10.1109/TIT.2026.3665439. arXiv, journal.

We prove new bounds on the generalized trifference function, T(n, m), which is the maximum number of ternary vectors of length n such that for any three of them there are at least m coordinates where they pairwise differ. This is a special case of the so-called k-hash codes, and we improve the old upper bounds on them by generalizing various bounds from the theory of error-correcting codes. We also compute the exact values of T(n, m) and its linear variant for small parameters. 

[24] Covering half-grids with lines and planes, with Shantanu Nene. arXiv.

We prove some tight bounds on hyperplane covering problems on `half-grids’, which are finite sets of points obtained by cutting a grid in half. 

[23] Explicit constructions of optimal blocking sets and minimal codes, with István Tomon, Combinatorica 46, 13 (2026). arXiv, journal. 

Building up on [21], we give explicit constructions of strong s-blocking sets and s-minimal codes for any fixed s that are optimal up-to a constant factor. We use certain tree like substructures of hypergraphs derived from expander graphs defined on points in `general position’. We also explore some connections to the size-Ramsey numbers of hypergraphs. 

[22] Ramsey numbers and extremal structures in polar spaces, with John Bamberg, Ferdinand Ihringer and Ananthakrishnan Ravi. arXiv.

We establish new connections between partial m-ovoids in finite classical polar spaces and Ramsey numbers. By using triangle-free graphs whose complement graphs have a low 2-rank, we give explicit constructions of large partial 2-ovoids in binary symplectic spaces. We also improve the lower bounds on the recently introduced rank-Ramsey problem and give new constructions of nearly k-orthogonal sets over the binary field.

[21] Strong blocking sets and minimal codes from expander graphs, with Noga Alon, Shagnik Das, and Alessandro Neri. Transactions of the AMS, 377, 5389-5410 (2024). arXiv. journal.

Using constant-degree expander graphs and finite geometry, we give the first explicit construction of minimal codes over \mathbb{F}_q that has dimension k and length at most cqk for some absolute constant c. Our construction relies on a graph parameter called vertex integrity, and it is the first of its kind in finite geometry. Another application is an explicit construction of an affine blocking set w.r.t. codimension-2 subspaces that is a constant factor away from the lower bound.

[20] Blocking sets, minimal codes, and trifferent codes, with Jozefien D’haeseleer, Dion Gijswijt and Aditya Potukuchi. Journal of the London Mathematical Society, 109, e12938 (2024) . arXiv. journal.

We establish connections and equivalences between affine blocking sets, strong blocking sets, minimal codes, and trifferent codes. Using these, we improve various bounds and give new explicit constructions. Our work gives a new method for attacking the decades old trifference problem.

[19] Covering grids with multiplicity, with Simona Boyadzhiyska, Shagnik Das, and Yvonne den Bakker, Combinatorial Theory, 3 #4 (2023). arXiv. journal.

We determine the minimum number of lines needed to cover every point of a grid at least k times while avoiding one of them, for various types of grids in the plane. In particular, we prove for most grids, the Ball-Serra lower bound can only be tight if the grid is “wide”.

[18] A new upper bound on the minimum degree of minimal Ramsey graphs, with Thomas Lesgourgues, Australasian Journal of Combinatorics, 92 65–69 (2025). arXiv. journal.

Using our methods from [14] and an old finite geometric construction, we improve the total degree of the polynomial upper bound on the minimum degree of minimal Ramsey graphs for cliques. We also show the optimality of this construction in some sense if one uses the packing of triangle-free partial linear spaces, which has been the only approach so far.

[17] Restricted Variable Chevalley-Warning Theorems, with Pete L. Clark. arXiv.

We prove some further generalisations of the Chevalley-Warning theorems.

[16] On the minimum degree of minimal Ramsey graphs for cliques versus cycles, with Simona Boyadzhiyska, Dennis Clemens, Pranshu Gupta, Thomas Lesgourgues and Anita Liebenau. SIAM Journal of Discrete Mathematics, 37 (2023). arXiv. journal.

We study the minimum degree of minimal Ramsey graphs for various asymmetric collections of graphs involving cycles and cliques.

[15] Subspace coverings with multiplicities, with Simona Boyadzhiyska, Shagnik Das and Tamás Mészarós. Combinatorics, Probability and Computing, 32 782–795 (2023). arXiv, blog, journal.

We study the problem of covering points of \mathbb{F}_2^n using codimension d affine subspaces such that the origin is covered at most k - 1 times while every other point is covered at least k times. We obtain exact results when either k or n is very large with respect to the other parameter, while establishing some interesting connections with coding theory.

[14] The minimum degree of minimal Ramsey graphs for cliques, with John Bamberg and Thomas Lesgourgues. Bulletins of London Math Society, 54, 1827-1838 (2022). arXiv, journal, blog.

Using a group theoretical model of generalized quadrangles, due to Kantor from 1980’s, we improve the general upper bound on a Ramsey parameter introduced by Burr, Erdős and Lovász in 1976.

[13] Ryser’s conjecture for t-intersecting hypergraphs, with Shagnik Das, Patrick Morris and Tibor Szabó. Journal of Combinatorial Theory, Series A. Volume 179 (2021), 105366, arXiv, journal, blog.

We improve known results for the problem of bounding the minimum cover number of an r-partite r-uniform t-intersecting hypergraph, and for t > r/3 we obtain sharp bounds. We also introduce various variants of Ryser’s conjecture and prove some initial results for them.

[12] A construction for clique-free pseudorandom graphs, with Ferdinand Ihringer and Valentina Pepe. Combinatorica 40 (2020), 307-314. arXiv, journal.

We construct a new family of optimally pseudorandom graphs that have no subgraphs isomorphic to a clique of fixed size k, and have higher edge density than the previous best construction of Alon and Krivelevich. Even a minor improvement to our construction will improve the best known lower bounds for off-diagonal Ramsey numbers, using the work of Mubayi and Verstraete. 

[11] Non-intersecting Ryser hypergraphs, with Valentina Pepe. SIAM J. Discrete Math. 34 (2020), 230–240. arXiv, journal.

Ryser’s conjecture is a longstanding open problem in combinatorics, which states that every r-partite, r-uniform hypergraph has a vertex cover of size at most (r - 1) times the matching number of the hypergraph. For r = 2, this is just Konig’s theorem, and besides that value of r the conjecture is only known to be true for r = 3. In recent years, extremal hypergraphs with respect to this conjecture, known as Ryser hypergraphs, have been studied extensively. We contribute to their study by giving new infinite families of non-intersecting Ryser hypergraphs, for uniformity at least 4, that cannot be built by taking disjoint copies of intersecting Ryser hypergraphs (and possibly adding some new edges), as was known to be true for 3-uniform hypergraphs. In this sense, it is the first truly non-intersecting infinite family of Ryser hypergraphs.

[10] A generalization of the theorems of Chevalley-Warning and Ax-Katz via polynomial substitutions. With Ioulia N. Baoulina and Pete L. Clark. Proceedings of the AMS 147 (2019), 4107–4122. arXiv, journal.

We prove a new generalization of the classical Chevalley-Warning theorem, Ax-Katz theorem and several related results.

[9] On regular induced subgraphs of generalized polygons. With John Bamberg and Gordon Royle. Journal of Combinatorial Theory, Series A. Volume 158 (2018), 254–275. arXiv, journal.

The cage problem asks for the smallest graph with a given degree k and girth g. The number of vertices in the smallest graph is denoted by c(k, g), and graphs meeting this bound are called cages. The only known non-trivial infinite families of cages are the generalized polygons, which tell us that c(k, g) = 2((k-1)^{g/2 - 1} + (k-1)^{g/2 - 2} + \cdots + 1) for g \in \{6, 8, 12\}, whenever k - 1 is a prime power. We improve the known upper bounds on c(k, 8) and c(k, 12) for infinitely many values of k by constructing new small regular induced subgraphs of known generalized quadrangles and hexagons. We also give a lower bound on the smallest number of vertices in a regular induced subgraph of a generalized polygon, using the expander mixing lemma, which gives a limit to our methods in improving the upper bounds on cage numbers.

[8] Minimal multiple blocking sets, with Sam Mattheus and Jeroen Schillewaert. Electronic Journal of Combinatorics. Volume 25, Issue 4 (2018),#P4.66. arxiv, journal.

Using the expander mixing lemma, we prove the first non-trivial upper bound on the size of a minimal t-fold blocking set in a finite projective plane. This generalises a classical result of Bruen and Thas. The techniques used also give new proofs of some old and new results in finite geometry. We talk about the sharpness of our result, some constructions, and generalizations.

[7] The \mathrm{L}_3(4) near octagon, with Bart De Bruyn. Journal of Algebraic Combinatorics. Volume 48, Issue 1 (2018), 157–178. arxiv, journal.

We give an alternate direct construction of one of the near octagons discovered in [2] using the projective special linear group \mathrm{PSL}_3(4). This construction is used to derive geometric and group theoretic properties of this near octagon, and we propose a new family of near octagons to which both this \mathrm{L}_3(4) near octagon and the \mathrm{G}_2(4) near octagon discovered in [2] belong. So far, these are the only two nontrivial members of this family that we know (we define and classify the trivial ones in the paper).

[6] On generalized hexagons of order (3, t) and (4, t) containing a subhexagon, with Bart De Bruyn. European Journal of Combinatorics.Volume 62 (2017), 115–123. arXiv, journal, blog.

We extend the work done in [1] by proving that there is no semi-finite hexagon containing any of the known generalized hexagons of order (3, 3) and (4, 4) as full subgeometry. Moreover, we show that the split Cayley hexagon \mathrm{H}(4) is not contained in any generalized hexagon as a full subgeometry. The code in main.g constructs computer models of small generalized hexagons that we use in our computations and the code in main.sage performs all the computations mentioned in Section 4.

[5] Some non-existence results for distance-j ovoids in small generalized polygons, with Ferdinand Ihringer. arXiv, journal (the journal version is much shorter)

We show that the dual split Cayley generalized hexagon \mathrm{H}(4)^D does not have any distance-2 ovoids (which are equivalent to exact hitting sets in the corresponding 5-regular 5-uniform hypergraph on 1365 vertices), and that the Ree-Tits octagon \mathrm{GO}(2, 4) does not have any distance-3 ovoids, thus resolving the last remaining case for existence of distance-3 ovoids in known finite generalized octagons. The computational techniques we use, which combine Knuth’s Dancing Links, Linton’s SmallestImageSet, Integer Linear Programming, along with a nice trick involving subgeometries, might be useful in small cases of other finite geometrical problems that have a similar flavour. Our complete computer code is available here.

[4] Characterizations of the Suzuki tower near polygons, with Bart De Bruyn. Designs, Codes and Cryptography. Volume 84, Issue 1 (2017), 115–133. arXiv, journal.

We prove uniqueness results for the near polygons lying in the Suzuki tower, that we described in [2]. In particular, we prove a characterization of the Hall-Janko near octagon as the unique near octagon of order (2, 4) containing the dual split Cayley hexagon \mathrm{H}(2)^D as a subgeometry. The following computer codes construct Table 1, 2, 3 and 4, and verify Lemmas 4.12, 5.1, 5.2, 5.3, 5.4: Suz1.g, Suz2.g, HallJankoHyp.g. Also see this for an independent verification by Bart.

[3] On Zeros of a Polynomial in a Finite Grid, with Pete L. Clark, Aditya Potukuchi and John R. Schmitt. Combinatorics, Probability and Computing. Volume 27, Issue 3 (2018), 310–333. arXiv, journal.

We prove a generalization of a result of Alon and Füredi, and show how this elementary result on zeros of polynomials is connected to various other results in Coding Theory, Finite Geometry and Theoretical Computer Science. This in combination with the earlier work of Clark, Forrow and Schmitt suggest that much like Combinatorial Nullstellensatz, the Alon-Füredi Theorem is a fundamental result on polynomials with connections to various important topics in mathematics. Here is a video of a talk I gave on this paper to a general scientific audience.

[2] A new near octagon and the Suzuki tower, with Bart De Bruyn. Electronic Journal of Combinatorics. Volume 23, Issue 2 (2016), #P2.35. arXiv, journal.

Bart and I discovered a new near octagon corresponding to the finite simple group G_2(4) in July 2014. Here we give its construction, prove several of its properties, and find a “Suzuki tower of near polygons” corresponding to the Suzuki tower of finite simple groups. We also give geometric constructions of some well known strongly regular graphs using this new near octagon and its subgeometries. Moreover, we construct another new near octagon as a subgeometry, related to the group L_3(4).

[1] On semi-finite hexagons of order (2, t) containing a subhexagon, with Bart De Bruyn. Annals of Combinatorics. Volume 20, Issue 3 (2016), 433–452. arXiv, journal.

Inspired by a famous open problem posed by Jacques Tits on existence of semi-finite generalized polygons, for which no progress has been made so far in the case of generalized hexagons, we solve a specific case of an easier version of the problem where non-existence is proved after assuming that the generalized hexagon contains a particular subhexagon. We show that (a) no generalized hexagon contains H(2) as a full proper subgeometry, (b) every near hexagon containing H(2)^D is a finite generalized hexagon, and hence isomorphic to H(2)^D or the triality hexagon T(8, 2)^D via the classification result of Cohen and Tits.
The following computer code constructs Table 1 and 2, and verifies Lemma 3.1: GSplit2.g.

Miscellaneous 

Finite geometry paves the way for breakthroughs in Ramsey theory, Nieuw Archief voor Wiskunde, Dec 2023.

Talks

  1. On semi-finite hexagons of order (2,t) containing a subhexagon of order 2, Fourth Irsee Conference (2014), slides.
  2. The Alon-Füredi bound, British Combinatorial Conference 2015, slides.
  3. Computing hyperplanes of near polygons, COCOA 2015, slides.
  4. On zeros of a polynomial in a finite grid: the Alon-Füredi bound, Combinatorics 2016 and at Discrete Mathematics Days 2016, slides.
  5. Zeros of polynomials over a finite grid, polynomial method workshop in HUJI organised by Jordan Ellenberg and Gil Kalai, 2016.
  6. Zeros of polynomials over a finite grid, Caltech combinatorics seminar and UCLA combinatorics seminar, 2017.
  7. Expander mixing lemma in finite geometry, UWA combinatorics seminar, 2017.
  8. The cage problem and finite geometry, Fq13, 2017.
  9. Minimal multiple blocking sets, Irsee, 2017, slides.
  10. Extremal problems in finite geometry, BMS-BGSMath Junior Meeting 2017, Institut d’Estudis Catalans, Barcelona, slides.
  11. Improved bounds on cage numbers, Combinatorics 2018, Facets of Complexity (colloquium talk), and the Berlin-Poznań-Hamburg Seminar.
  12. Extremal and near extremal constructions for Ryser’s conjecture, 1st Southwestern German Workshop on Graph Theory, 2018.
  13. Extremal problems in finite geometry, Humboldt Network Meeting, Leipzig, 2019.
  14. Clique-free pseudorandom graphs, Finite Geometry and Extremal Combinatorics conference, University of Delaware, 2019.
  15. Clique-free pseudorandom graphs, 42nd ACCMCC, 2019.
  16. The Chevalley-Warning theorems, UWA combinatorics seminar, 2020.
  17. Clique-free pseudorandom graphs, TU Delft optimization seminar, 2020.
  18. What is Ramsey Theory?, TU Delft Lunch Colloquium, Nov 2020.
  19. Lower bounds for diagonal Ramsey numbers, FU Berlin Combinatorics seminar, Nov 2020.
  20. Subspace coverings with multiplicities, UvA general mathematics colloquium, Feb 2021.
  21. The minimum degree of minimal Ramsey graphs for cliques, Georgia tech combinatorics seminar, Feb 2021.
  22. The minimum degree of minimal Ramsey graphs for cliques, Combinatorics 2022, Mantua, May-June 2022.
  23. Trifferent codes and affine blocking sets, Algebraic and Geometric methods in Coding Theory (ACA 2022 Special Session in Istanbul-Gebze), August 2022.
  24. Affine blocking sets and trifferent codes, Aart and Combinatorics, Nesin Mathematics Village, Izmir, Turkey, February 2023.
  25. Codes, blocking sets and graphs, Dutch Days of Combinatorics, Utrecht, Netherlands, March 2023.
  26. Blocking sets and minimal codes from expander graphs, 10th Slovenian Conference on Graph Theory, June 2023.
  27. Determining Ramsey numbers using finite geometry, DMO seminar, TU Delft, June 2023.
  28. Blocking sets, minimal codes, and trifferent codes, SIAM Conference on Applied Algebraic Geometry (AG23) – Minisymposium on coding theory and Galois geometry, July 2023.
  29. Expander graphs, strong blocking sets and minimal codes, Eurocomb, Prague, August 2023.
  30. Grid Covering Problems, NYC Geometry Seminar, September 2023. Recording
  31. Covering finite grids, Finite Geometry Workshop Szeged, October 2023
  32. Covering grids with multiplicities, Summit280, Budapest, Hungary, July 2024.
  33. Oddtowns, partial ovoids, and the rank-Ramsey problem, August 2024, Sustech, Shenzhen, China.
  34. Linear trifferent codes, 2024 Colloquia in Combinatorics, London, May 2024
  35. Grid covering problems, Coding Theory Colloquium, Ghent University, November 2024 
  36. Oddtowns, nearly orthogonal sets, and the rank-Ramsey problem, TUM, Munich, Germany April 2025.
  37. Covering problems on finite grids, Umeå Discrete Mathematics Seminar, May 2025.
  38. Using expander graphs to construct strong blocking sets and minimal codes, Finite Geometry and Extremal Combinatorics Workshop, IASM-BIRS, Hangzhou, June 2025.
  39. Using expander graphs to construct strong blocking sets and minimal codes, Finite Geometry, Algebraic Combinatorics, and Coding Theory e-seminars, July 2025.
  40. What is mathematics research?, Kaveri Group of Institutes, Pune, October 2025.
  41. The chromatic number of finite projective spaces, IECM-2026, Pune, January 2026.
  42. Circular sorting, DCU-UCD Discrete Mathematics Seminar, Dublin, February 2026.

Collaborators

Noga Alon (1), John Bamberg (3), Paul Bastide (1), Ioulia N. Baoulina (1), Simona Boyadzhiyska (3), Wouter Cames van Batenburg (1), Pete L. Clark (3), Dennis Clemens (1), Shagnik Das (4), Jozefien D’haeseleer (1), Bart De Bruyn (5), Yvonne den Bakker (1), Dion Gijswijt (2), Carla Groenland (1), Pranshu Gupta (1), Ferdinand Ihringer (3), Rohinee Joshi (1), Bartłomiej Kielak (1), Benedek Kovács (1), Thomas Lesgourgues (3), Zoltán Lóránt Nagy (1), Anita Liebanau (1), Sam Mattheus (1), Tamás Mészarós (1), Patrick Morris (1), Shantanu Nene (1), Alessandro Neri (1), Valentina Pepe (2), Aditya Potukuchi (2), Ananthakrishnan Ravi (2), Gordon Royle (1), Jeroen Schillewaert (1), John R. Schmitt (1), Gábor Somlai (1), Tibor Szabó (1), István Tomon (1), Máté Vizer (1), Zeyu Zheng (1).

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