30 citations · 90 across the 27 of their papers we have counts for
41 papers
Fluctuations of Nonlinear Observables in Mean Field Neural Network Training
Arnaud Descours, Geoffrey Lacour
Mean field limits describe the training dynamics of wide neural networks through the evolution of the empirical distribution of their parameters. Although functional central limit…
A General Upper Bound on Multicolor Ordered Ramsey Numbers
Martin Balko, Klára Grinerová
We provide a general upper bound on multicolor ordered Ramsey numbers in terms of the interval chromatic number and the degeneracy of an ordered graph. We extend previous results b…
Bolzano: Case Studies in LLM-Assisted Mathematical Research
Martin Balko, Jan Grebík, Pavel Hubáček +5
We report new results on eight problems in mathematics and theoretical computer science, produced with the assistance of Bolzano, an open-source multi-agent LLM system. Bolzano orc…
Massively Parallel Proof-Number Search for Impartial Games and Beyond
Tomáš Čížek, Martin Balko, Martin Schmid
Proof-Number Search is a best-first search algorithm with many successful applications, especially in game solving. As large-scale computing clusters become increasingly accessible…
Crossing and non-crossing families
Todor Antić, Martin Balko, Birgit Vogtenhuber
For a finite set of points in the plane in general position, a \emph{crossing family} of size in is a collection of line segments with endpoints in that are pai…
Faces in rectilinear drawings of complete graphs
Martin Balko, Anna Brötzner, Fabian Klute +1
We initiate the study of extremal problems about faces in convex rectilinear drawings of~, that is, drawings where vertices are represented by points in the plane in convex po…
A Survey on Ordered Ramsey Numbers
Martin Balko
The ordered Ramsey number of a graph with a linearly ordered vertex set is the smallest positive integer such that any two-coloring of the edges of the ordered complete g…
On forbidden configurations in point-line incidence graphs
Martin Balko, Nóra Frankl
The celebrated Szemerédi--Trotter theorem states that the maximum number of incidences between points and lines in the plane is , which is asymptotically tight.…
On the Uncrossed Number of Graphs
Martin Balko, Petr Hliněný, Tomáš Masařík +3
Visualizing a graph in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masařík and Hliněný [GD 2023] recently a…
Ramsey theorem for trees with successor operation
Martin Balko, David Chodounský, Natasha Dobrinen +4
We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for reg…
On ordered Ramsey numbers of matchings versus triangles
Martin Balko, Marian Poljak
For graphs and with linearly ordered vertex sets, the \ordered Ramsey number is the smallest positive integer such that any red-blue coloring of the…
Characterisation of the big Ramsey degrees of the generic partial order
Martin Balko, David Chodounský, Natasha Dobrinen +4
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite p…
On Helly numbers of exponential lattices
Gergely Ambrus, Martin Balko, Nóra Frankl +2
Given a set , define the \emph{Helly number of }, denoted by , as the smallest positive integer , if it exists, for which the following statem…
On ordered Ramsey numbers of tripartite 3-uniform hypergraphs
Martin Balko, Máté Vizer
For an integer , an ordered -uniform hypergraph is a -uniform hypergraph together with a fixed linear ordering of its vertex set. The or…
The constant of point-line incidence constructions
Martin Balko, Adam Sheffer, Ruiwen Tang
We study a lower bound for the constant of the Szemerédi-Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies $I({\mathcal P…
Bounding and computing obstacle numbers of graphs
Martin Balko, Steven Chaplick, Robert Ganian +4
An obstacle representation of a graph consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of to points such that…
Erdős--Szekeres-type problems in the real projective plane
Martin Balko, Manfred Scheucher, Pavel Valtr
We consider point sets in the real projective plane and explore variants of classical extremal problems about planar point sets in this setting, with a main focus o…
On off-diagonal ordered Ramsey numbers of nested matchings
Martin Balko, Marian Poljak
For two graphs and with linearly ordered vertex sets, the ordered Ramsey number is the minimum such that every red-blue coloring of the edges of the…
Tight bounds on the expected number of holes in random point sets
Martin Balko, Manfred Scheucher, Pavel Valtr
For integers and , a -hole in a set of points in general position in is a -tuple of points from in convex position such that the…
Exact big Ramsey degrees for finitely constrained binary free amalgamation classes
Martin Balko, David Chodounský, Natasha Dobrinen +4
We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recen…
Implementation of Sprouts: a graph drawing game
Tomáš Čížek, Martin Balko
Sprouts is a two-player pencil-and-paper game invented by John Conway and Michael Paterson in 1967. In the game, the players take turns in joining dots by curves according to simpl…
Big Ramsey degrees and forbidden cycles
Martin Balko, David Chodounský, Jan Hubička +3
Using the Carlson-Simpson theorem, we give a new general condition for a structure in a finite binary relational language to have finite big Ramsey degrees
Big Ramsey degrees of the generic partial order
Martin Balko, David Chodounský, Natasha Dobrinen +4
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the ge…
Big Ramsey degrees of 3-uniform hypergraphs are finite
Martin Balko, David Chodounský, Jan Hubička +2
We prove that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees. This is the first case where big Ramsey degrees are known to be finite for structures in…