Schedule and Abstracts 2024/2025

Date Speaker Title
22.01.2025
10:15 am
A3/SR024 		Jack Allsop (Monash University) Subsquares of Latin squares
16.01.2025 Dingding Dong (Harvard University) Uncommon linear systems of two equations
09.01.2025 Mehmet Akif Yildiz (University of Amsterdam) Path decompositions of oriented graphs
18.12.2024 Christoph Spiegel (Zuse-Institut Berlin) Coloring the plane with neural networks
04.12.2024 Milos Stojakovic (University of Novi Sad) Maker-Breaker games on random boards
28.11.2024 Christoph Spiegel (Zuse-Institut Berlin) An Unsure Talk on an Un-Schur Problem
14/21.11.2024 Aldo Kiem (Zuse-Institut Berlin) Forcing Graphs and Graph Algebra Operators
21.10.2024 Victor Falgas-Ravry (Umea University) 1-independent percolation on high dimensional lattices
17.10.2024 Joseph Fehribach (Worcester Polytechnic Institute) Families of Kirchhoff Graphs

Abstracts

22.01.2025
Jack Allsop (Monash University)
Subsquares of Latin squares
Abstract: A Latin square of order \(n\) is an \(n \times n\) matrix of \(n\) symbols, each of which occurs exactly once in each row and column. A subsquare of a Latin square is a submatrix which is itself a Latin square. Any Latin square of order \(n\) has \(n^2\) subsquares of order \(1\) and a single subsquare of order \(n\). Any other subsquare is called proper. In this talk we discuss a number of problems regarding subsquares. Given integers \(n\) and \(m\), what is the maximum number of subsquares of order \(m\) in a Latin square of order \(n\)? What is the expected number of subsquares of order \(m\) in a uniformly random Latin square of order \(n\)? For what orders do there exist Latin squares without any proper subsquares?
16.01.2025
Dingding Dong (Harvard University)
Uncommon linear systems of two equations
Abstract: A system of linear equations \(L\) is common over \(F_p\) if any 2-coloring of \(F_p^n\) gives at least as many monochromatic solutions to it as a random 2-coloring, asymptotically as \(n\to\infty\). It is an open question which linear systems are common. When \(L\) is a single equation, Fox, Pham and Zhao gave a complete characterization of common linear equations. When \(L\) consists of two equations, Kamčev, Liebenau and Morrison showed that all irredundant 2*4 linear systems are uncommon. In joint work with Anqi Li and Yufei Zhao, we: (1) determine commonness of all 2*5 linear systems up to a small number of cases; (2) show that all \(2*k\) linear systems with \(k\) even and girth (length of the shortest equation) \(k-1\) are uncommon, answering a question of Kamčev-Liebenau-Morrison.
09.01.2025
Mehmet Akif Yildiz (University of Amsterdam)
Path decompositions of oriented graphs
Abstract: We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths in any path decomposition dictated by vertex degree imbalances, and any directed graph that meets this bound is called consistent.
In 1976, Alspach, Mason, and Pullman conjectured that (as a generalization of Kelly's conjecture on Hamilton decompositions of regular tournaments) every tournament of even order is consistent. This conjecture was recently verified for large tournaments by Girão, Granet, Kühn, Lo, and Osthus. A stronger conjecture, proposed by Pullman in 1980, states that every orientation of every regular graph with odd degree is consistent.
In this talk, I will present our recent work (joint with Viresh Patel) that establishes Pullman’s conjecture for random regular graphs and regular graphs without short cycles.
18.12.2024
Christoph Spiegel (Zuse-Institut Berlin)
Coloring the plane with neural networks
Abstract: We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance din the sixth color. Such colorings have previously been known to exist for \(0.41 < \sqrt 2−1 \le d \le 1/\sqrt 5 < 0.45\). Our results significantly expand that range to \(0.354 \le d \le0.657\), the first improvement in 30 years. The constructions underlying this notably were derived by formalizing colorings suggested by a custom machine learning approach. This is joint work with Sebastian Pokutta, Konrad Mundiger, and Max Zimmer.
04.12.2024
Milos Stojakovic (University of Novi Sad)
Maker-Breaker games on random boards
Abstract: In Maker-Breaker games played on edge sets of graphs, two players, Maker and Breaker, alternately claim unclaimed edges of a given graph until all of its edges are claimed. Maker wins the game if he claims all edges of one representative of a prescribed graph-theoretic structure (e.g. a Hamiltonian cycle, or a fixed graph \(H\)). Breaker wins otherwise.
We take a closer look at various Maker-Breaker games played on the edge sets of random graphs.
28.11.2024
Christoph Spiegel (Zuse-Institut Berlin)
An Unsure Talk on an Un-Schur Problem
Abstract: Graham, Rödl, and Ruciński originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first \(n\) integers. This question was subsequently resolved independently by Datskovsky, Schoen, and Robertson and Zeilberger. In this talk we will study a natural anti-Ramsey variant of this question and establish the first non-trivial bounds by proving that the maximum fraction of Schur triples that can be rainbow in a given \(3\)-coloring of the first \(n\) integers is at least \(0.4\) and at most \(0.66656\). We conjecture the lower bound to be tight. This question is also motivated by a famous analogous problem in graph theory due to Erdős and Sós regarding the maximum number of rainbow triangles in any \(3\)-coloring of \(K_n\), which was settled by Balogh et al. The contents of this talk are joint work with Olaf Parczyk.
14/21.11.2024
Aldo Kiem (Zuse-Institut Berlin)
Forcing Graphs and Graph Algebra Operators
Abstract: We report on recent progress on Sidorenko's conjecture and the forcing conjecture. Our main observation is that graph operators such as subdivisions, blow-ups or adding disjoint vertices correspond to order-preserving operators on Graph Algebras. This way we easily obtain some new results, in particular that the proper balanced blow-ups of Sidorenko graphs are forcing. This is joint work with Olaf Parczyk and Christoph Spiegel. The talk will survey our results and assume no prerequisites with graph algebras and categories.
21.10.2024
Victor Falgas-Ravry (Umea University)
1-independent percolation on high dimensional lattices
Abstract: The celebrated Harris-Kesten theorem states that if each edge of the square integer lattice is open with probability \(p\), independently of all other edges, then for \(p\) at most \(1/2\) almost surely all connected components of open edges are finite, while if \(p>1/2\) almost surely there exists a unique infinite connected component of open edges.
What if one allows the state of an edge (open or closed) to depend on that of neighbouring edges? Could one use such local dependencies to delay the emergence of an infinite component? Such questions lead to fascinating problems at the interface between extremal combinatorics and percolation theory. In this talk, I will present recent work related to one such decade-old and still open problem of Balister and Bollobás on 1-independent percolation on high-dimensional integer lattices.
Joint work with Vincent Pfenninger (TU Graz)
17.10.2024
Joseph Fehribach (Worcester Polytechnic Institute)
Families of Kirchhoff Graphs
Abstract: Given a set of \(n\) vectors in any vector space over the rationals, suppose that \(k<n\) are linear independent. Kirchhoff graphs are vector graphs (graphs whose edges are these vectors), whose cycles represent the dependencies of these vectors and whose vertex cuts are orthogonal to these cycles. This presentation discusses the uniformity of vector 2-connected Kirchhoff graphs, and how graph tiling can generate families of Kirchhoff graphs. These families are composed of prime graphs (those having no Kirchhoff subgraphs), and composite graphs (not prime), all generated by a set of fundamental Kirchhoff graphs (smallest prime Kirchhoff graphs that can generate other prime and composite graphs for our set of vectors).

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