The research of the group focuses on interrelated aspects of combinatorics: Graph theory, Random Graphs, Probabilistic method, Geometric group theory and algebraic methods, Enumerative combinatorics, Combinatorial geometry, and Combinatorial number theory. Some recent research achievements are: Proof of the Maximum Distance Separable codes conjecture for prime fields; solution of a conjecture of Green on the removal lemma for systems of equations in finite fields

### Recent Submissions

• #### Degrees of compression and inertia for free-abelian times free groups ﻿

(Elsevier, 2021-02-15)
Article
Restricted access - publisher's policy
We introduce the concepts of degree of inertia, diG(H), and degree of compression,dcG(H), of a finitely generated subgroupHof a given groupG. For the case of direct productsof free-abelian and free groups, we compute the ...
• #### A removal lemma for systems of linear equations over finite fields ﻿

(2012-01)
Article
Open Access
We prove a removal lemma for systems of linear equations over finite fields: let X1, . . . , Xm be subsets of the finite field Fq and let A be a (k × m) matrix with coefficients in Fq and rank k; if the linear system Ax = ...
• #### Triangulations and a discrete Brunn–Minkowski inequality in the plane ﻿

(2019-08-29)
Article
Open Access
For a set A of points in the plane, not all collinear, we denote by tr(A) the number of triangles in a triangulation of A, that is, tr(A)=2i+b-2 , where b and i are the numbers of boundary and interior points of the convex ...
• #### Product of primes in arithmetic progressions ﻿

(2020-05-01)
Article
Open Access
We prove that, for all q=2 and for all invertible residue classes a modulo q, there exists a natural number n=(650q)9 that is congruent to a modulo q and that is the product of exactly three primes, all of which are below ...
• #### Frozen (¿ + 1)-colourings of bounded degree graphs ﻿

(2020-10-19)
Article
Restricted access - publisher's policy
Let G be a graph on n vertices and with maximum degree ¿, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at ...
• #### On a problem of Sárközy and Sós for multivariate linear forms ﻿

(2020-03-18)
Article
Open Access
We prove that for pairwise co-prime numbers k1,…,kd=2 there does not exist any infinite set of positive integers A such that the representation function rA(n)=# {(a1,…,ad)¿Ad:k1a1+¿ +kdad=n} becomes constant for n large ...
• #### An Erdös–Fuchs theorem for ordered representation functions ﻿

(2020-10-28)
Article
Open Access
Letk=2 be a positive integer. We study concentration results for the ordered representationfunctionsr=k(A, n) = #{(a1= ··· =ak)¿ Ak:a1+···+ak=n}andr<k(A, n) = #{(a1<···<ak)¿ Ak:a1+···+ak=n}for any infinite set ...
• #### Local convergence and stability of tight bridge-addable classes ﻿

Article
Restricted access - publisher's policy
A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, ...
• #### Structure and enumeration of K4-minor-free links and link-diagrams ﻿

(2020-10-01)
Article
Open Access
We study the class of link-types that admit a -minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of . We prove that is the closure of a subclass of ...
• #### On the number of coloured triangulations of d-manifolds ﻿

(2020-03-11)
Article
Open Access
We give superexponential lower and upper bounds on the number of coloured d-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and d=3 is fixed. In the ...
• #### On varieties defined by large sets of quadrics and their application to error-correcting codes ﻿

(2020-10-01)
Article
Restricted access - publisher's policy
Let U be a ( k-1 2 - 1)-dimensional subspace of quadratic forms defined on F k with the property that U does not contain any reducible quadratic form. Let V (U) be the points of PG(k - 1, F) which are zeros of all quadratic ...
• #### The Glauber dynamics for edge-colorings of trees ﻿

(2020-09-26)
Article
Restricted access - publisher's policy
Let T be a tree on n vertices and with maximum degree ¿. We show that for k = ¿ + 1 the Glauber dynamics for k-edge-colourings of T mixes in polynomial time in n. The bound on the number of colours is best possible as the ...