We show that the average length of a fundamental cycle with respect to any fixed spanning tree of the n× n square grid is at least Ω(log n); the bound is asymptotically tight. This result answers in the affirmative a question posed by McCarty in relation to sparse representations of binary matroids.

The notion of first order convergence of graphs unifies the notions of convergence for sparse and dense graphs. Nešetřil and Ossona de Mendez [J. Symbolic Logic 84 (2019), 452-472] proved that every first order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit and conjectured the existence of such modeling limits with an additional property, the strong finitary mass transport principle. The existence of modeling limits satisfying the strong finitary mass transport principle was proved for first order convergent sequences of trees by Nešetřil and Ossona de Mendez [Electron. J. Combin. 23 (2016), P2.52] and for first order sequences of graphs with bounded path-width by Gajarský et al. [Random Structures Algorithms 50 (2017), 612-635]. We establish the existence of modeling limits satisfying the strong finitary mass transport principle for first order convergent sequences of graphs with bounded tree-width.

A graph H is common if its Ramsey multiplicity, i.e., the minimum number of monochromatic copies of H contained in any 2-edge-coloring of Kn, is asymptotically the same as the number of monochromatic copies in the random 2-edge-coloring of Kn. ErdoH\s conjectured that every complete graph is common, which was disproved by Thomason in the 1980s. Until today, a classification of common graphs remains a wide open and challenging problem. Grzesik et al. [Combin. Probab. Comput., 31 (2022), 907--923] conjectured that every k-apex of any connected Sidorenko graph is common. We prove for k \leq 5 that the k-apex of any tree is common.

The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.

Turán's Tetrahedron Problem asks to determine the Turán density of the complete hypergraph K_4^(3) (tetrahedron). This problem, posed by Turán in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract $500 prize from Erdős. In the 1980s, Erdős and Sós asked to determine Turán densities of K_4^(3)- (broken tetrahedron) and K_4^(3) (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Turán density of K_4^(3) is equal to 1/2; this confirms that Rödl's lower bound construction from 1986 is optimal.

We give a complete characterization of tournaments H that have the Sidorenko property with respect to nearly regular tournaments, i.e., the homomorphism density of H among all nearly regular tournaments is minimized by a random tournament. Corollaries of our result are a positive answer to the question of Noel, Ranganathan and Simbaqueba whether there exist infinitely many non-transitive tournaments that are quasirandom forcing for nearly regular tournaments, and a negative answer to their question whether almost every tournament is quasirandom forcing for nearly regular tournaments.

Erd\H os, Lov\'asz and Spencer showed in the late 1970s that the dimension of the region of $k$-vertex graph profiles, i.e., the region of feasible densities of $k$-vertex graphs in large graphs, is equal to the number of non-trivial connected graphs with at most $k$ vertices. We determine the dimension of the region of $k$-vertex tournament profiles. Our result, which explores an interesting connection to Lyndon words, yields that the dimension is much larger than just the number of strongly connected tournaments, which would be the answer expected as the analogy to the setting of graphs.

The contraction$^*$-depth is the matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph $G$ is the minimum height of a rooted forest whose closure contains $G$ by proving the following for every matroid $M$ (except the trivial case when $M$ consists of loops and bridges only): the contraction$^*$-depth of $M$ plus one is equal to the minimum contraction-depth of a matroid containing $M$ as a restriction.

In the 1980s, Erdős and Sós initiated the study of Turán hypergraph problems with a uniformity condition on the distribution of edges, i.e., determining density thresholds for the existence of a hypergraph H in a host hypergraph with edges uniformly distributed. In particular, Erdős and Sós asked to determine the uniform Turán densities of the hypergraphs K_4^(3)- and K_4^(3). After more than 30 years, the former was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open. In these two cases and several additional cases, the tight lower bounds are provided by a so-called palette construction. Lamaison [arXiv:2408.09643] has recently showed that the uniform Turán density of a 3-uniform hypergraph H is equal to the supremum of the densities of palettes that H is not colorable with. We give a necessary and sufficient condition, which is easy to verify, on the existence of a 3-uniform hypergraph colorable by a set of palettes and not colorable by another given set of palettes. We also demonstrate how our result can be used to prove the existence of 3-uniform hypergraphs with specific values of the uniform Turán density.

A classical result of Erd\H os, Lov\'asz and Spencer from the late 1970s asserts that the dimension of the feasible region of homomorphic densities of graphs with at most $k$ vertices in large graphs is equal to the number of connected graphs with at most $k$ vertices. Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of $k$-patterns is at least the number of non-trivial indecomposable permutations of size at most $k$. We identify a larger set of permutations, which are called Lyndon permutations, whose pattern densities are independent, and show that the dimension of the feasible region of densities of $k$-patterns is equal to the number of non-trivial Lyndon permutations of size at most $k$.

We construct a planar graph with twin-width equal to seven.

The workshop provided a venue for discussing several recent major developments in graph theory, with a primary focus on results concerning the notion of twin-width and hereditary properties of graphs. Both areas have seen a very rapid development recently, as reflected in the many results presented during the workshop. In addition to many interesting talks spanning the whole breadth of graph theory, the workshop also offered valuable collaboration opportunities, which also engaged early career researchers.

In the 1980s, Erd & odblac;s and S & oacute;s initiated the study of Tur & aacute;n problems with a uniformity condition on the distribution of edges: the uniform Tur & aacute;n density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least $d$ contains $H$ . In particular, they asked to determine the uniform Tur & aacute;n densities of $K_4<^>{(3)-}$ and $K_4<^>{(3)}$ . After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349 - 366] and [J. Eur. Math. Soc. 20 (2018), 1139 - 1159], while the latter still remains open. Till today, there are known constructions of $3$ -uniform hypergraphs with uniform Tur & aacute;n density equal to $0$ , $1/27$ , $4/27$ , and $1/4$ only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Tur & aacute;n density to be equal to $8/27$ and identify hypergraphs satisfying this condition.

Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of K_k, and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko's Conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common, and in fact, no 2-chromatic common graph unsettled for Sidorenko's Conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known. We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami, Hladky, Kral, Norine and Razborov [Combin. Probab. Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov [London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger, Stovicek and Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common graph with chromatic number at least four.

We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.

Hutchinson, Richter and Seymour [J. Combin. Theory Ser. B 84 (2002), 225-239] showed that every Eulerian triangulation of an orientable surface that has a sufficiently high representativity is 4-colorable. We give an explicit bound on the representativity in the case of the torus by proving that every Eulerian triangulation of the torus with representativity at least 10 is 4-colorable. We also observe that the bound on the representativity cannot be decreased to less than 8 as there exists a non-4-colorable Eulerian triangulation of the torus with representativity 7.

A combinatorial object is said to be quasirandom if it exhibits certain properties that are typically seen in a truly random object of the same kind. It is known that a permutation is quasirandom if and only if the pattern density of each of the twenty-four 4-point permutations is close to 1/24, which is its expected value in a random permutation. In other words, the set of all twenty-four 4-point permutations is quasirandom-forcing. Moreover, it is known that there exist sets of eight 4-point permutations that are also quasirandom-forcing. Breaking the barrier of linear dependency of perturbation gradients, we show that every quasirandom-forcing set of 4-point permutations must have cardinality at least five.

We study generalized quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lov\'asz and S\'os showed that the structure of such graphs is forced by homomorphism densities of graphs with at most $(10q)^q+q$ vertices; subsequently, Lov\'asz refined the argument to show that graphs with $4(2q+3)^8$ vertices suffice. Our results imply that the structure of generalized quasirandom graphs with $q\ge 2$ parts is forced by homomorphism densities of graphs with at most $4q^2-q$ vertices, and, if vertices in distinct parts have distinct degrees, then $2q+1$ vertices suffice. The latter improves the bound of $8q-4$ due to Spencer.

The notion of branch-depth for matroids was introduced by DeVos, Kwon and Oum as the matroid analogue of the tree-depth of graphs. The contraction-deletion-depth, another tree-depth like parameter of matroids, is the number of recursive steps needed to decompose a matroid by contractions and deletions to single elements. Any matroid with contraction-deletion-depth at most d has branch-depth at most d. However, the two notions are not functionally equivalent as contraction-deletion-depth of matroids with branch-depth two can be arbitrarily large. We show that the two notions are functionally equivalent for representable matroids when minor closures are considered. Namely, an F-representable matroid has small branch-depth if and only if it is a minor of an F-representable matroid with small contraction-deletion-depth. This implies that any class of F-representable matroids has bounded branch-depth if and only if it is a subclass of the minor closure of a class of F-representable matroids with bounded contraction-deletion-depth.

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ_1-norm of the Graver basis is bounded by a function of the maximum ℓ_1-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the ℓ_1-norm of the Graver basis of the constraint matrix, when parameterized by the ℓ_1-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.