The Erdős-Sós Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta \gt 0$ and $k_0\in \mathbb N$ such that the conjecture holds for every tree $T$ with $k \ge k_0$ edges and every graph $G$ with $|V(G)| \le (1+\delta )|V(T)|$.

The Erdős–Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős–Simonovits type stability theorem in multi-partite graphs. Different from the Erdős–Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$-free graph $G$ is close to the extremal graphs for $H$, then $G$ has a well-defined structure but may be far away from the extremal graphs for $H$. As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique.

We consider the hypergraph Turán problem of determining $ex(n, S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on $n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of $S^d$) as a subcomplex. We show that if there is an affirmative answer to a question of Gromov about sphere enumeration in high dimensions, then $ex(n, S^d) \geq \Omega (n^{d + 1 - (d + 1)/(2^{d + 1} - 2)})$. Furthermore, this lower bound holds unconditionally for 2-LC (locally constructible) spheres, which includes all shellable spheres and therefore all polytopes. We also prove an upper bound on $ex(n, S^d)$ of $O(n^{d + 1 - 1/2^{d - 1}})$ using a simple induction argument. We conjecture that the upper bound can be improved to match the conditional lower bound.

The Kruskal–Friedman theorem asserts: in any infinite sequence of finite trees with ordinal labels, some tree can be embedded into a later one, by an embedding that respects a certain gap condition. This strengthening of the original Kruskal theorem has been proved by I. Kříž (Ann. Math. 1989), in confirmation of a conjecture due to H. Friedman, who had established the result for finitely many labels. It provides one of the strongest mathematical examples for the independence phenomenon from Gödel’s theorems. The gap condition is particularly relevant due to its connection with the graph minor theorem of N. Robertson and P. Seymour. In the present article, we consider a uniform version of the Kruskal–Friedman theorem, which extends the result from trees to general recursive data types. Our main theorem shows that this uniform version is equivalent both to $\Pi ^1_1$-transfinite recursion and to a minimal bad sequence principle of Kříž, over the base theory $\mathsf {RCA_0}$ from reverse mathematics. This sheds new light on the role of infinity in finite combinatorics.

In this article, we study the algebra of Veronese type. We show that the presentation ideal of this algebra has an initial ideal whose Alexander dual has linear quotients. As an application, we explicitly obtain the Castelnuovo–Mumford regularity of the Veronese type algebra. Furthermore, we give an effective upper bound on the multiplicity of this algebra.

Modified ascent sequences, initially defined as the bijective images of ascent sequences under a certain hat map, have also been characterized as Cayley permutations where each entry is a leftmost copy if and only if it is an ascent top. These sequences play a significant role in the study of Fishburn structures. In this paper, we investigate (primitive) modified ascent sequences avoiding a pattern of length 4 by combining combinatorial and algebraic techniques, including the application of the kernel method. Our results confirm several conjectures posed by Cerbai.

Let $\Gamma $ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq \Gamma $, a domatic $\aleph _0$-partition (for its Schreier graph on $\Gamma $) is a partial function $f:\Gamma \rightharpoonup \mathbb {N}$ such that for every $x\in \Gamma $, one has $f[S\cdot x]=\mathbb {N}$. We show that a continuous domatic $\aleph _0$-partition exists, if and only if a Baire measurable domatic $\aleph _0$-partition exists, if and only if the topological closure of S is uncountable. A Haar measurable domatic $\aleph _0$-partition exists for all choices of S. We also investigate domatic partitions in the general descriptive graph combinatorial setting.

We generalize Baker–Bowler’s theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets and orthogonal vector sets, and establish basic properties on functoriality, duality and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we give a new proof that an orthogonal matroid is regular if and only if it is representable over ${\mathbb F}_2$ and ${\mathbb F}_3$, which was originally shown by Geelen [16], and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over ${\mathbb F}_3$ and ${\mathbb F}_4$.

For an ideal I in a Noetherian ring R, the Fitting ideals $\mathrm{Fitt}_j(I)$ are studied. We discuss the question of when $\mathrm{Fitt}_j(I)=I$ or $\sqrt{\mathrm{Fitt}_j(I)}=\sqrt{I}$ for some j. A classical case is the Hilbert–Burch theorem when $j=1$ and I is a perfect ideal of grade 2 in a local ring.

The lattice walks in the plane starting at the origin $\mathbf {0}$ with steps in $\{-1,0,1\}^{2}\setminus \{\mathbf {0}\}$ are called king walks. We investigate enumeration and divisibility for higher dimensional king walks confined to certain regions. Specifically, we establish an explicit formula for the number of $(r+s)$-dimensional king walks of length n ending at $(a_1,\ldots ,a_r,b_1,\ldots ,b_s)$ which never dip below $x_i=0$ for $i=1,\ldots ,r$. We also derive divisibility properties for the number of $(r+s)$-dimensional king walks of length p (an odd prime) through group actions.

Given a symmetric monoidal category ${\mathcal C}$ with product $\sqcup $, where the neutral element for the product is an initial object, we consider the poset of $\sqcup $-complemented subobjects of a given object X. When this poset has finite height, we define decompositions and partial decompositions of X which are coherent with $\sqcup $, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness.

In well-studied scenarios, the poset of $\sqcup $-complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of nondegenerate subspaces of a vector space with a nondegenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.

Let $t\geq 2$ and $k\geq 1$ be integers. A t-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t. Let $b_{t,k}(n)$ denote the number of hooks of length k in all the t-regular partitions of n. In this article, we prove some inequalities for $b_{t,k}(n)$ for fixed values of k. We prove that for any $t\geq 2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq 0$. We also prove that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n>3$, and $b_{3,3}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. Finally, we state some problems for future works.

We consider the problem of sequential matching in a stochastic block model with several classes of nodes and generic compatibility constraints. When the probabilities of connections do not scale with the size of the graph, we show that under the Ncond condition, a simple max-weight type policy allows us to attain an asymptotically perfect matching while no sequential algorithm attains perfect matching otherwise. The proof relies on a specific Markovian representation of the dynamics associated with Lyapunov techniques.

Let ${\mathscr {G}} $ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb {C}$, excluding the absolutely special case of $A^{(2)}_{2\ell }$. Using the methods and results of Zhu, we prove a duality theorem for general ${\mathscr {G}} $: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for ${\mathscr {G}} $. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of ${\mathscr {G}} $. In particular, this confirms a conjecture of Haines and Richarz.

In this article, we investigate the possibility of generating all the configurations of a subshift in a local way. We propose two definitions of local generation, explore their properties and develop techniques to determine whether a subshift satisfies these definitions. We illustrate the results with several examples.

We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald P-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra.

Let $\mathcal {D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object R. Let $\Lambda =\operatorname {End}_{\mathcal {D}}R$ be the endomorphism algebra of R. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with the mutation of support $\tau $-tilting $\Lambda $-modules. In the case that $\mathcal {D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $\tau $-tilting $\Lambda $-modules is isomorphic to the graph of flips of certain collections of tagged arcs on the surface, which is moreover proved to be connected. Consequently, the mutation graph of support $\tau $-tilting modules over a skew-gentle algebra is connected. This generalizes one main result in [49].

The Jacobian of a very general complex algebraic curve of genus at least 3 contains an algebraic cycle called the Ceresa cycle that is homologically trivial but algebraically nontrivial. Zharkov defined in analogy the tropical Ceresa cycle of a metric graph and proved a similar result for very general tropical curves overlying the complete graph on four vertices. We extend this result by considering a related, ‘universal’ invariant of the underlying graph called the Ceresa period; we show that having trivial Ceresa period has a forbidden minor characterization that coincides with the graph being of hyperelliptic type.

We prove a criterion of when the dual character $\chi _{D}(x)$ of the flagged Weyl module associated a diagram D in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $\chi _{D}(x)$ are either 0 or 1. This settles a conjecture proposed by Mészáros–St. Dizier–Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink–Mészáros–St. Dizier and Hodges–Yong, respectively.

Let W be a simply laced Weyl group of finite type and rank n. If W has type $E_7$, $E_8$ or $D_n$ for n even, then the root system of W has subsystems of type $nA_1$. This gives rise to an irreducible Macdonald representation of W spanned by n-roots, which are products of n orthogonal roots in the symmetric algebra of the reflection representation. We prove that in these cases, the set of all maximal sets of orthogonal positive roots has the structure of a quasiparabolic set in the sense of Rains–Vazirani. The quasiparabolic structure can be described in terms of certain quadruples of orthogonal positive roots which we call crossings, nestings and alignments. This leads to nonnesting and noncrossing bases for the Macdonald representation, as well as some highly structured partially ordered sets. We use the $8$-roots in type $E_8$ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the $E_8$ root system.