Gábor Somlai
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Gábor Somlai has a total of 26 co-authors. Here are the top co-authors by number of shared publications:
- Gergely Kiss
Department of Mathematics Corvinus University of Budapest Budapest Hungary
9 shared publications - Máté Vizer
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS BUDAPEST HUNGARY
3 shared publications Benedek Kovács
ELTE Linear Hypergraphs Research Group, Etvs Lornd University, Budapest, Hungary
2 shared publications- Dávid Matolcsi
Eötvös Loránd University, Budapest, Hungary
2 shared publications - Gergely Kiss
Alfréd Rényi Institute of Mathematics, HUN-REN, Budapest, Hungary
2 shared publications - Máté Matolcsi
HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
2 shared publications
Latest Publications
- 16 February 2026
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 72 (5) , 2907-2914
The Generalized Trifference Problem
We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. This problem is a special case of the perfect $k$ -hashing problem in theoretical computer science, corresponding to the $k=3$ case. For $m = 1$ , we get the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$ . By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters. Moreover, we relate the trifference problem to the sunflower conjecture.
- 16 February 2026
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 72 (5) , 2907-2914
The Generalized Trifference Problem
We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. This problem is a special case of the perfect $k$ -hashing problem in theoretical computer science, corresponding to the $k=3$ case. For $m = 1$ , we get the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$ . By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters. Moreover, we relate the trifference problem to the sunflower conjecture.
- 15 January 2026
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 154 (3) , 1073-1076
Spectrality of the product does not imply that the components are spectral
Greenfeld and Lev [J. Anal. Math., 16 (2020), 409–441] conjectured that the Cartesian product of two sets A A and B B is spectral if and only if A A and B B are spectral. We construct a counterexample to this conjecture using the existence of a tile that has no spectra.
- 15 January 2026
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 154 (3) , 1073-1076
Spectrality of the product does not imply that the components are spectral
Greenfeld and Lev [J. Anal. Math., 16 (2020), 409–441] conjectured that the Cartesian product of two sets A A and B B is spectral if and only if A A and B B are spectral. We construct a counterexample to this conjecture using the existence of a tile that has no spectra.
- Open Access
Dihedral groups of square-free order are DCI-groups
A finite group $G$ is a called a DCI-group if any two isomorphic Cayley digraphs of $G$ are also isomorphic via an automorphism of $G$. If $G$ is a non-abelian generalised dihedral DCI-group, then Dobson, Muzychuk, and Spiga proved that $G$ must be a dihedral group of square-free order (Ars Math. Contemp., 2022). In this paper, we prove that the converse statement also holds, i.\,e., all dihedral groups of square-free order are DCI-groups.Open AccessDihedral groups of square-free order are DCI-groups
A finite group $G$ is a called a DCI-group if any two isomorphic Cayley digraphs of $G$ are also isomorphic via an automorphism of $G$. If $G$ is a non-abelian generalised dihedral DCI-group, then Dobson, Muzychuk, and Spiga proved that $G$ must be a dihedral group of square-free order (Ars Math. Contemp., 2022). In this paper, we prove that the converse statement also holds, i.\,e., all dihedral groups of square-free order are DCI-groups. - Open Access
The generalized trifference problem
We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.Open AccessThe generalized trifference problem
We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters. - Open Access
- 29 April 2024
- Published by Springer Nature in Designs, Codes and Cryptography
- Vol. 92 (9) , 2587-2597
Special directions on the finite affine plane
In this paper we study the number of special directions of sets of cardinality divisible by p on a finite plane of order p, where p is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly three special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that have exactly four special directions for small values of p.
- 29 April 2024
- Published by Springer Nature in Designs, Codes and Cryptography
- Vol. 92 (9) , 2587-2597
Open AccessSpecial directions on the finite affine plane
In this paper we study the number of special directions of sets of cardinality divisible by p on a finite plane of order p, where p is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly three special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that have exactly four special directions for small values of p.
- Open Access
- 24 April 2024
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 261 (2) , 1-52
On the diameter of Cayley graphs of classical groups with generating sets containing a transvection
- 24 April 2024
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 261 (2) , 1-52
Open AccessOn the diameter of Cayley graphs of classical groups with generating sets containing a transvection
- Open Access
- 8 April 2024
- Published by Springer Nature in Archiv der Mathematik
- Vol. 122 (6) , 575-580
A new proof of Rédei’s theorem on the number of directions
Rédei and Megyesi proved that the number of directions determined by a p-element subset of $${\mathbb F}_p^2$$ is either 1 or at least $$\frac{p+3}{2}$$ . The same result was independently obtained by Dress, Klin, and Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields.- 8 April 2024
- Published by Springer Nature in Archiv der Mathematik
- Vol. 122 (6) , 575-580
Open AccessA new proof of Rédei’s theorem on the number of directions
Rédei and Megyesi proved that the number of directions determined by a p-element subset of $${\mathbb F}_p^2$$ is either 1 or at least $$\frac{p+3}{2}$$ . The same result was independently obtained by Dress, Klin, and Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields. - Open Access
- 4 December 2023
- Published by Springer Nature in Sampling Theory, Signal Processing, and Data Analysis
- Vol. 22 (1) , 1-12
Tiling and weak tiling in $(\mathbb {Z}_p)^d$We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups $$(\mathbb {Z}_p)^d$$ , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $$d=1, 2$$ , and prove some partial results for $$d=3$$ .- 4 December 2023
- Published by Springer Nature in Sampling Theory, Signal Processing, and Data Analysis
- Vol. 22 (1) , 1-12
Open AccessTiling and weak tiling in $(\mathbb {Z}_p)^d$We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups $$(\mathbb {Z}_p)^d$$ , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $$d=1, 2$$ , and prove some partial results for $$d=3$$ . - 5 October 2023
- Published by World Scientific Pub Co Pte Ltd in International Journal of Foundations of Computer Science
- Vol. 34 (07) , 737-760
Optimal Embedded and Enclosing Isosceles Triangles
Given a triangle [ ], we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of [ ] with respect to area and perimeter. This problem was initially posed by Nandakumar [17, 22] and was first studied by Kiss, Pach, and Somlai [13], who showed that if [ ] is the smallest area isosceles triangle containing [ ], then [ ] and [ ] share a side and an angle. In the present paper, we prove that for any triangle [ ], every maximum area isosceles triangle embedded in [ ] and every maximum perimeter isosceles triangle embedded in [ ] shares a side and an angle with [ ]. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles [ ] whose minimum perimeter isosceles containers do not share a side and an angle with [ ].
- 5 October 2023
- Published by World Scientific Pub Co Pte Ltd in International Journal of Foundations of Computer Science
- Vol. 34 (07) , 737-760
Optimal Embedded and Enclosing Isosceles Triangles
Given a triangle [ ], we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of [ ] with respect to area and perimeter. This problem was initially posed by Nandakumar [17, 22] and was first studied by Kiss, Pach, and Somlai [13], who showed that if [ ] is the smallest area isosceles triangle containing [ ], then [ ] and [ ] share a side and an angle. In the present paper, we prove that for any triangle [ ], every maximum area isosceles triangle embedded in [ ] and every maximum perimeter isosceles triangle embedded in [ ] shares a side and an angle with [ ]. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles [ ] whose minimum perimeter isosceles containers do not share a side and an angle with [ ].
- Open Access
Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication
The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the possible decompositions of the transcendental extension of the rational field of rank one into two pieces. Further, we prove that the positive elements of a real algebraic extensions of the rational numbers are indecomposable into two pieces.
Open AccessDecompositions of the positive real numbers into disjoint sets closed under addition and multiplication
The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the possible decompositions of the transcendental extension of the rational field of rank one into two pieces. Further, we prove that the positive elements of a real algebraic extensions of the rational numbers are indecomposable into two pieces.