A bipartite graph B is called a brace if it is connected and every matching of size at most two in B is contained in some perfect matching of B. A conformal cross over some cycle C is a pair of disjoint paths P_1 , P_2 which are internally disjoint from C, the endpoints of each path separate the endpoints of the other path on C, and both C∪ P_1∪ P_2 and B-(V(C)∪ V(P_1)∪ V(P_2)) have a perfect matching. We show that if C is a 4-cycle in a brace B, then C has a a conformal cross if and only if B contains K_3,3 as a matching minor. This result implies a polynomial time algorithm which solves the 2-linkage problem for alternating paths in bipartite graphs with perfect matchings.
The modern theory of homogeneous structures begins with the work of Roland Fraïssé. The theory developed in the last seventy years is placed in the border area between combinatorics, model theory, algebra, and analysis. We turn our attention to its combinatorial pillar, namely, the work on the classification of structures for given homogeneity types, and focus onto the homomorphism homogeneous ones, introduced in 2006 by Cameron and Nešetřil. An oriented graph is called homomorphism homogeneous if every homomorphism between finite induced subgraphs extends to an endomorphism. In this paper we present a complete classification of the countable homomorphism homogeneous oriented graphs. Among these we identify those that are polymorphism homogeneous. Here an oriented graph is called polymorphism homogeneous if each of its finite powers is homomorphism homogeneous.
This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set A containing Ω(|A|^3/2) three-term arithmetic progressions.
We show that if the ground set of a matroid can be partitioned into k≥ 2 bases, then for any given subset S of the ground set, there is a partition into k bases such that the sizes of the intersections of the bases with S may differ by at most one. This settles the matroid equitability conjecture by Fekete and Szabó (Electron. J. Comb. 2011) in the affirmative. We also investigate equitable splittings of two disjoint sets S_1 and S_2 , and show that there is a partition into k bases such that the sizes of the intersections with S_1 may differ by at most one and the sizes of the intersections with S_2 may differ by at most two; this is the best one can hope for arbitrary matroids. We also derive applications of this result to matroid-constrained fair division problems. We show that there exists a matroid-constrained allocation that is envy-free up to one item if the valuations are identical and tri-valued additive. We also show that for bi-valued additive valuations, there exists a matroid-constrained allocation that provides everyone their maximin share.
Abstract polytopes generalize the face lattice of convex polytopes. A polytope is semiregular if its facets are regular and its automorphism group acts transitively on its vertices. In this paper we construct semiregular, facet-transitive polyhedra with trivial facet stabilizer, showing that semiregular abstract polyhedra can have an unbounded number of flag orbits, while having as little as one facet orbit. We interpret this construction in terms of operations applied to high rank regular and chiral polytopes, and we see how these same operations help us construct alternating semiregular polyhedra (that is, with two facet orbits and adjacent facets in different orbits). Finally, we give an idea to generalize this construction giving examples in higher ranks.
A graph G=(V,E) is geometrically embeddable into a normed space X when there is a mapping ζ :V→ X such that ‖ζ (v)-ζ (w)‖ _X⩽ 1 if and only if {v,w}∈ E , for all distinct v,w∈ V . Our result is the following universal threshold for the embeddability of trees. Let Δ⩾ 3 , and let N be sufficiently large in terms of Δ . Every N–vertex tree of maximal degree at most Δ is embeddable into any normed space of dimension at least 64 log N/loglog N , and complete trees are non-embeddable into any normed space of dimension less than 1/2 log N/loglog N . In striking contrast, spectral expanders and random graphs are known to be non-embeddable in sublogarithmic dimension. Our result is based on a randomized embedding whose analysis utilizes the recent breakthroughs on Bourgain’s slicing problem.
Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 years. In this paper, we study the following Ramsey version of these problems. Given a set L⊆{0,…,k-1} and a family ℱ of k-element sets which does not contain a sunflower with m petals whose kernel size is in L, how large a subfamily of ℱ can we find in which no pair has intersection size in L? We give matching upper and lower bounds, determining the dependence on m for all k and L. This problem also finds applications in quantum computing. As an application of our techniques, we also obtain a variant of Füredi's celebrated semilattice lemma, which is a key tool in the powerful delta-system method. We prove that one cannot remove the double-exponential dependency on the uniformity in Füredi's result, however, we provide an alternative with significantly better, single-exponential dependency on the parameters, which is still strong enough for most applications of the delta-system method.
An n-vertex graph is degree 3-critical if it has 2n - 2 edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erdős, Faudree, Gyárfás, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szabó through a construction based on leaf-to-leaf paths in trees whose vertices have degree either 1 or 3. They went on to suggest several weaker conjectures about cycle lengths in degree 3-critical graphs and leaf-to-leaf path lengths in these so-called 1-3 trees. We resolve three of their questions either fully or up to a constant factor. Our main results are the following: Several of our proofs rely on purely combinatorial means, while others exploit a connection to an additive problem that might be of independent interest.
We characterize all orientations of cycles C for which for every fixed ε > 0 there exists a constant c ≥ 1 such that every digraph D without loops or parallel arcs with χ(D) ≥ c and minimum out-degree at least ε |V(D)| contains C as a subdigraph. This generalizes a result of Thomassen.
Classical results of Cauchy [4] and Dehn [5] imply that the 1-skeleton of a convex simplicial polyhedron P is rigid, i.e. every continuous motion of the vertices of P in ℝ^3 which preserves its edge lengths results in a polyhedron which is congruent to P. This result was extended to convex simplicial polytopes in ℝ^d for all d≥ 3 by Whiteley [16], and to generic realisations of 1-skeletons of simplicial (d-1) -manifolds in ℝ^d by Kalai [8] for d≥ 4 and Fogelsanger [6] for d≥ 3 . We will generalise Kalai’s result by showing that, for all d≥ 4 and any fixed 1≤ k≤ d-3 , every generic realisation of the k-skeleton of a simplicial (d-1) -manifold in ℝ^d is volume rigid, i.e. every continuous motion of its vertices in ℝ^d which preserves the volumes of its k-faces results in a congruent realisation. In addition, we conjecture that our result remains true for k=d-2 and verify this conjecture when d=4,5,6 .
Consider the process where the n vertices of a square 2-dimensional torus appear consecutively in a random order. We show that typically the size of the 3-core of the corresponding induced unit-distance graph transitions from 0 to n-o(n) within a single step. Equivalently, by infecting the vertices of the torus in a random order, under two-neighbour bootstrap percolation, the size of the infected set transitions instantaneously from o(n) to n. This hitting time result answers a question of Benjamini. We also study the much more challenging and general setting of bootstrap percolation on two-dimensional lattices for a variety of finite-range infection rules. In this case, powerful but fragile bootstrap percolation tools such as the rectangles process and the Aizenman–Lebowitz lemma become unavailable. We develop a new method complementing and replacing these standard techniques, thus allowing us to prove the above hitting time result for a wide family of threshold bootstrap percolation rules on the 2-dimensional square lattice, including neighbourhoods given by large ℓ ^p balls for p∈ [1,∞ ] .
Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted k-th powers. They conjectured that for each k≥ 2, if one changes o(X^1/k) elements of M_k'={x^k+1: x ∈ℕ} up to X, then the resulting set cannot be written as a product set AB nontrivially. In this paper, we confirm a more general version of their conjecture for k≥ 3.
A strong s-blocking set in a projective space is a set of points that intersects each codimension-s subspace in a spanning set of the subspace. We present an explicit construction of such sets in a (k - 1)-dimensional projective space over 𝔽_q of size O_s(q^s k), which is optimal up to the constant factor depending on s. This also yields an optimal explicit construction of affine blocking sets in 𝔽_q^k with respect to codimension-(s+1) affine subspaces, and of s-minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong 1-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong s-blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.
We characterize the obstructions to the Erdős-Pósa property of A-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group Γ and for every subset Λ of Γ , the family of Γ -labelled A-paths whose lengths are in Λ satisfies the half-integral Erdős-Pósa property. Moreover, we give a characterization of such Γ and Λ⊆Γ for which the same family of A-paths satisfies the full Erdős-Pósa property.
We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in ℂ^n , as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in ℂ^n is equal to 2^n+1-1 and that the fractional illumination number of the polydisc in ℂ^n is equal to 2^n . In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.
Recently, Zhang and Wu proved a conjecture of Kalai and Meshulam, showing that for every graph G without induced cycles of length divisible by 3, the sum of all reduced Betti numbers of its independence complex I(G) is at most 1. We extend this result to the hypergraph setting. Namely, we show that the same conclusion holds for any hypergraph H that does not contain a Berge cycle of length divisible by 3. This establishes a broader connection between forbidden cycle structures and the topological simplicity of independence complexes. As a key tool, we introduce a hypergraph analogue of Barmak's star cluster theorem for graphs. This new theorem implies, in particular, that if a hypergraph H has a vertex v that is not isolated and is not contained in an induced Berge cycle of length 3, then there exists a hypergraph H' with fewer vertices than H such that the independence complex of H is homotopy equivalent to the suspension of the independence complex of H'.
One of the foundational theorems of extremal graph theory is Dirac’s theorem, which says that if an n-vertex graph G has minimum degree at least n/2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work by Sárközy, Selkow and Szemerédi showed that in fact Dirac graphs have many Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph G (in terms of an entropy-like parameter of G). In this paper we extend Cuckler and Kahn’s result to perfect matchings in hypergraphs. For positive integers d0 ), then the number of perfect matchings in G is controlled by an entropy-like parameter of G. This strengthens cruder estimates arising from work of Kang–Kelly–Kühn–Osthus–Pfenninger and Pham–Sah–Sawhney–Simkin.
The hypergraph container lemma is a powerful tool in probabilistic combinatorics that has found many applications since it was first proved a decade ago. Roughly speaking, it asserts that the family of independent sets of every uniform hypergraph can be covered by a small number of almost-independent sets, called containers. In this article, we formulate and prove two new versions of the lemma that display the following three attractive features. First, they both admit short proofs that have surprising connections to other well-studied topics in probabilistic combinatorics. Second, they use alternative notions of almost-independence in order to describe the containers. Third, they yield improved dependence of the number of containers on the uniformity of the hypergraph, hitting a natural barrier for second-moment-type approaches.
We study colorful no-dimensional Tverberg-type problems and obtain several optimal results. A colorful no-dimensional Tverberg-type theorem provides a bound on a radius R such that, for any pairwise disjoint k-element subsets Q_1,…,Q_n of a normed space, there exists a partition of Q_1∪⋯∪ Q_n into disjoint transversals {P_1,…,P_k} for which a ball of radius R intersects the convex hull of each P_i (1≤ i≤ k). Our methods are deterministic and dimension-free, and they are unified by optimizing two functionals: a quadratic selection functional whose local maximizers produce a complete system of disjoint transversals, and a convex intersection functional that certifies a common point. First, in the Euclidean setting we bound R in terms of the Chebyshev radii (minimal enclosing-ball radii) of the color classes Q_1,…,Q_n. A key observation is a “combinatorial” subadditivity of the squared Chebyshev radius: given sequences X=(x_1,…,x_k) and Y=(y_1,…,y_k) of points in a Euclidean space, contained in balls of radii R_X and R_Y (not necessarily with the same center), one can reenumerate Y so that the pointwise-sum sequence Z=(x_1+y_1,…,x_k+y_k) is contained in a ball of radius R_Z satisfying R_Z^2 ≤ R_X^2 + R_Y^2 . As a corollary, we obtain the best-possible bound R ≤1/√(2n)√(k-1/k) max_1≤ i≤ ndiam(Q_i). Our algorithm returns the desired disjoint transversals in overall time 𝒪(nk^3). Second, we develop a complementary approach based on the inter-color diameter and extend the framework to obtain no-dimensional colorful Tverberg-type results in the hyperbolic setting and in Banach spaces.
The Haglund–Haiman–Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: H̃_μ(X;q,t)=∑ _σ : μ→ℙx^σt^(σ )q^(σ ). Inspired by Martin’s multiline-queue formula for the stationary distribution of multitype asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams recently introduced the queue inversion statistic and conjectured that the tableaux formula for H̃_μ(X;q,t) is invariant if the inversion statistic is replaced by . This was subsequently resolved by Ayyer, Mandelshtam and Martin, who proposed a stronger conjecture on the equivalence of the two refined formulas for H̃_μ(X;q,t) . Our main result confirms this Ayyer–Mandelshtam–Martin conjecture. We establish an equidistribution between the pairs (,) and (,) of μ -Mahonian statistics on any row-equivalency class [τ ] , where τ is a filling of the Young diagram of μ . As a byproduct of our approach, we show that if τ is a rectangular filling, the triples (,,) and (,,) have the same distribution over [τ ] .
