We prove that a wide range of coloring problems for graphs on surfaces can be resolved by inspecting a finite number of configurations.

We prove log-concavity of exit probabilities of lattice random walks in certain planar regions.

Let 2[n] and (\(\matrix{{\left[ n \right]} \cr i \cr } \)) be the power set and the collection of all i-subsets of {1, 2, …, n}, respectively. We call t (t ≥ 2) families \({{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq {2^{\left[ n \right]}}\) cross-intersecting if Ai ∩ Aj ≠ ∅ for any \({A_i} \in {{\cal A}_i}\) and \({A_j} \in {{\cal A}_j}\) with i ≠ j. We show that, for n ≥ k +l, l ≥ r

For n ∈ ℕ, let h(n) denote the number of simplicial complexes on n vertices up to homotopy equivalence. Here we prove that \(h(n)\geq 2^{2^{0.02n}}\) when n is large enough. Together with the trivial upper bound of \(2^{2^{n}}\) on the number of labeled simplicial complexes on n vertices this proves a conjecture of Kalai that h(n) is doubly exponential in n.

In the (1:b) component game played on a graph G, two players, Maker and Breaker, alternately claim 1 and b previously unclaimed edges of G, respectively. Maker’s aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty (b

We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: one is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs.

The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size \(\cal{O}(k\log k)\). The analogous result for directed graphs has been proven by Reed, Robertson, Seymour, and Thomas, but their proof yields a nonelementary dependency of the size of

We prove that for every positive integer m there is a finite point set \(\cal{P}\) in the plane such that no matter how \(\cal{P}\) is three-colored, there is always a disk containing exactly m points, all of the same color. This improves a result of Pach, Tardos and Tóth who proved the same for two colors. The main ingredient of the construction is a subconstruction whose points are in convex position

The following generalisation of the Erdős Unit Distance problem was recently suggested by Palsson, Senger, and Sheffer. For a fixed sequence δ = (δ1, …, δk) of k distances, a (k + 1)-tuple (p1, …, pk+1) of distinct points in ℝd is called a k-chain if ∥pj − pj+1∥ = δj for every 1 ≤ j ≤ k. What is the maximum number C dk (n) of k-chains in a set of n points in ℝd? Improving the results of Palsson, Senger

Is there a universal constant K, say K = 3, such that one may dispose of all pairs of adjacent vertices with equal degrees from any given connected graph of order at least three by blowing its selected edges into at most K parallel edges? This question was first posed in 2004 by Karoński, Łuczak and Thomason, who equivalently asked if one may assign weights 1,2,3 to the edges of every such graph so

We resolve the Ramsey problem for {x,y,z: x + y = p(z)} for all polynomials p over ℤ. In particular, we characterise all polynomials that are 2-Ramsey, that is, those p(z) such that any 2-colouring of ℕ contains infinitely many monochromatic solutions for x+y=p(z). For polynomials that are not 2-Ramsey, we characterise all 2-colourings of ℕ that are not 2-Ramsey, revealing that certain divisibility

Kim, Kühn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655–4742) greatly extended the well-known blow-up lemma of Komlós, Sárközy and Szemerédi by proving a ‘blow-up lemma for approximate decompositions’ which states that multipartite quasirandom graphs can be almost decomposed into any collection of bounded degree graphs with the same multipartite structure and slightly fewer edges. This

Van der Holst and Pendavingh introduced a graph parameter σ, which coincides with the more famous Colin de Verdière graph parameter μ for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G) for any graph G. We confirm this conjecture. As far as we know, this is

In 1985, Mader conjectured that for every acyclic digraph F there exists K = K(F) such that every digraph D with minimum out-degree at least K contains a subdivision of F. This conjecture remains widely open, even for digraphs F on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases of Mader’s problem and made the following conjecture: for every ℓ ≥ 2 there

A Kempe switch of a 3-edge-coloring of a cubic graph G on a bicolored cycle C swaps the colors on C and gives rise to a new 3-edge-coloring of G. Two 3-edge-colorings of G are Kempe equivalent if they can be obtained from each other by a sequence of Kempe switches. Fisk proved that any two 3-edge-colorings in a cubic bipartite planar graph are Kempe equivalent. In this paper, we obtain an analog of

Form an n × n matrix by drawing entries independently from {±1} (or another fixed nontrivial finitely supported distribution in Z) and let φ be the characteristic polynomial. We show, conditionally on the extended Riemann hypothesis, that with high probability φ is irreducible and Gal(φ) ≥ An.

For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identity $${\rm{NC}}_{n + 1}^{(k)}(t) = t\sum\limits_{i = 0}^n {\left({\matrix{n \cr i \cr}} \right)} {\rm{NW}}_i^{(k)}(t),$$ where NC (k)m (t) (resp. NW (k)m (t)) is the sum of weights, tnumber of blocks, of partitions of {1,…,m} without k-crossings (resp. enhanced k-crossings). The special k = 2 and t =

We work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of homomorphism densities of finite trees and it is characterized in terms of distributions on iterated degree measures, Markov operators, weak isomorphism of a conditional expectation with respect to invariant sub-σ-algebras

We show that for any n ≥ 13, there exist graphs with chromatic number larger than n whose product has chromatic number n. Our construction is an adaptation of the construction of counterexamples to Hedetniemi’s conjecture devised by Shitov, and adapted by Zhu to graphs with relatively small chromatic numbers. The new tools we introduce are graphs with minimal colourings that are “wide” in the sense

Balanced weighing matrices with parameters $$\left({1 + 18 \cdot {{{9^{m + 1}} - 1} \over 8},{9^{m + 1}},4 \cdot {9^m}} \right),$$ for each nonzero integer m are constructed. This is the first infinite class not belonging to those with classical parameters. It is shown that any balanced weighing matrix is equivalent to a five-class association scheme.

Let Qn be the n-dimensional Hamming cube and N = 2n. We prove that the number of maximal independent sets in Qn is asymptotically $$2n{2^{N/4}},$$ as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an

Let V ⊆ [n] be a k-element subset of [n]. The uniform distribution on the 2k strings from {0, 1}n that are set to zero outside of V is called an (n, k)-zero-fixing source. An ϵ-extractor for (n, k)-zero-fixing sources is a mapping F: {0, 1}n → {0, 1}m, for some m, such that F(X) is ϵ-close in statistical distance to the uniform distribution on {0, 1}m for every (n, k)-zero-fixing source X. Zero-fixing

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c ≥ 13 and d ≥ 1, we give first explicit constructions of c-crossing-critical graphs containing arbitrarily many vertices of degree greater than d. We also show that such unbounded degree constructions do not exist for c ≤ 12, precisely

The book graph \(B_n^{(k)}\) consists of n copies of Kk+1 joined along a common Kk. The Ramsey numbers of \(B_n^{(k)}\) are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed k, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide

A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of Kn is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagger, Štovíček and Thomason [20]. We also show that a graph H is k-common for every k if and only if H is Sidorenko and that H is locally k-common for every

In 1959, Grötzsch [5] famously proved that every planar graph of girth at least 4 is 3-colourable (or equivalently, admits a homomorphism to C3). A natural generalization of this is the following conjecture: for every positive integer t, every planar graph of girth at least 4t admits a homomorphism to C2t+1. This is in fact the planar dual of a well-known conjecture of Jaeger [7] which states that

We study the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated polytope to cover the whole space. As our main result, we describe a new algorithm for this problem, which is simpler, more efficient and easier to implement than the only prior

A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the tetrahedron exhibits this phenomenon assuming Turán’s conjecture. Our main result is to construct a finite family of triple systems \({\cal M}\), determine its Turán

The orbifold Euler characteristic of the moduli space \({{\cal M}_{g;1}}\) of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: \(\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)\), where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics

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 a non-binary language. Our proof is based on the vector (or product) form of Milliken’s Tree Theorem and demonstrates a general method to carry existing results on structures in binary relational languages to higher arities

Let d be a positive integer and U ⊂ ℤd finite. We study $$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B + U} \right|} \over {{{\left| A \right|}^{1/2}}{{\left| B \right|}^{1/2}}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, ∣U + U∣/∣U∣. For instance, we show $$\beta (U) = \left| U

Let m be a positive integer, q be a prime power, and PG(2,q) be the projective plane over the finite field \({\mathbb{F}_q}\). Finding complete m-arcs in PG(2,q) of size less than q is a classical problem in finite geometry. In this paper we give a complete answer to this problem when q is relatively large compared with m, explicitly constructing the smallest m-arcs in the literature so far for any

A zero-one matrix M contains a zero-one matrix A if one can delete some rows and columns of M, and turn some 1-entries into 0-entries such that the resulting matrix is A. The extremal number of A, denoted by ex(n, A), is the maximum number of 1-entries in an n × n sized matrix M that does not contain A. A matrix A is column-t-partite (or row-t-partite), if it can be cut along the columns (or rows)

Gardner conjectured that if two bounded measurable sets A, B ⊂ ℝn are equidecomposable by a set of isometries Γ generating an amenable group then A and B admit a measurable equidecomposition by all isometries. Cieśla and Sabok asked if there is a measurable equidecomposition using isometries only in the group generated by Γ. We answer this question negatively.

We break the exponential barrier for triangulations of the real projective space, constructing a trianglation of ℝPn with \({e^{\left({{1 \over 2} + o(1)} \right)\sqrt n \log n}}\) vertices.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum ∑1/(n log n) for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that ∑n−λ over a primitive set is maximized by

In this note we investigate how to use an initial portion of the intersection array of a distance-regular graph to give an upper bound for the diameter of the graph. We prove three new diameter bounds. Our bounds are tight for the Hamming d-cube, doubled Odd graphs, the Heawood graph, Tutte’s 8-cage and 12-cage, the generalized dodecagons of order (1, 3) and (1, 4), the Biggs-Smith graph, the Pappus

We prove a sharp upper bound on the number of shortest cycles contained inside any connected graph in terms of its number of vertices, girth, and maximal degree. Equality holds only for Moore graphs, which gives a new characterization of these graphs. In the case of regular graphs, our result improves an inequality of Teo and Koh. We also show that a subsequence of the Ramanujan graphs of Lubotzky-Phillips-Sarnak

A ladder is a 2 × k grid graph. When does a graph class \({\cal C}\) exclude some ladder as a minor? We show that this is the case if and only if all graphs G in \({\cal C}\) admit a proper vertex coloring with a bounded number of colors such that for every 2-connected subgraph H of G, there is a color that appears exactly once in H. This type of vertex coloring is a relaxation of the notion of centered

One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graph on n vertices has at most ⌊n2/4⌋ edges. About half a century later Andrásfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on n vertices without independent sets of size αn, where 2/5 ≤ α < 1/2, are blow-ups of the pentagon. More than 50 further years have elapsed

In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with

We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear as common factors of two automatic sequences defined over multiplicatively independent bases. This generalises the famous theorem of Cobham.

In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over ℝ and ℂ, which can be considered metric versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.

We prove that the number of directions contained in a set of the form A × B ⊂ AG(2,p), where p is prime, is at least |A||B| − min{|A|, |B|} + 2. Here A and B are subsets of GF(p) each with at least two elements and |A||B|

Motivated by the work of Lovász and Szegedy on the convergence and limits of dense graph sequences [10], we investigate the convergence and limits of finite trees with respect to sampling in normalized distance. We introduce dendrons (a notion based on separable real trees) and show that the sampling limits of finite trees are exactly the dendrons. We also prove that the limit dendron is unique.

A cap of spherical radius α on a unit d-sphere S is the set of points within spherical distance α from a given point on the sphere. Let \({\cal F}\) be a finite set of caps lying on S. We prove that if no hyperplane through the center of S divides \({\cal F}\) into two non-empty subsets without intersecting any cap in \({\cal F}\), then there is a cap of radius equal to the sum of radii of all caps

We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborová and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation method from mathematical physics. In the case of random regular graphs the lower bound can be expressed algebraically, while in the case of the binomial random we

A step forward is made in a long standing Lovász problem regarding existence of Hamilton paths in vertex-transitive graphs. It is shown that a vertex-transitive graph of order a product of two primes arising from a primitive action of PSL(2;p) on the cosets of a subgroup isomorphic to Dp−1 has a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton

The asymptotic spectrum of graphs, introduced by Zuiddam (Combinatorica, 2019), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms between the complements. He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic

Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colorable. This is a strengthening of the famous Hadwiger’s Conjecture. Geelen et al. proved that every graph with no odd Kt minor is \(O(t\sqrt {\log t} )\)-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no Kt minor, we make the first improvement on this bound by

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively.

The main result of this paper is that, if Γ is a finite connected 4-valent vertex- and edge-transitive graph, then either Γ is part of a well-understood family of graphs, or every non-identity automorphism of Γ fixes at most 1/3 of the vertices. As a corollary, we get a similar result for 3-valent vertex-transitive graphs.

Pokrovskiy conjectured that there is a function f: ℕ → ℕ such that any 2k-strongly-connected tournament with minimum out and in-degree at least f(k) is k-linked. In this paper, we show that any (2k + 1)-strongly-connected tournament with minimum out-degree at least some polynomial in k is k-linked, thus resolving the conjecture up to the additive factor of 1 in the connectivity bound, but without the

Besides various asymptotic results on the concept of sum-product bases in the set of non-negative integers ℕ, we investigate by probabilistic arguments the existence of thin sets A, A′ of non-negative integers such that AA + A = ℕ and A′A′ + A′A′ = ℕ.

Let f be a smooth real function with strictly monotone first k derivatives. We show that for a finite set A, with ∣A + A∣ ≤K∣A∣, $$\left| {{2^k}f(A) - ({2^k} - 1)f(A)} \right|{ \gg _k}\,{\left| A \right|^{k + 1 - o(1)}}/{K^{{O_k}(1)}}.$$ We deduce several new sum-product type implications, e.g. that A+A being small implies unbounded growth for a many enough times iterated product set A ⋯ A.

We describe a family of graphs with queue-number at most 4 but unbounded stack-number. This resolves open problems of Heath, Leighton and Rosenberg (1992) and Blankenship and Oporowski (1999).

Let G be a finite group and let h be a positive integer. A BH(G, h) matrix is a G-invariant ∣G∣ × ∣G∣ matrix H whose entries are complex hth roots of unity such that H H* = ∣G∣I∣G∣, where H* denotes the complex conjugate transpose of H, and I∣G∣ denotes the identity matrix of order ∣G∣. In this paper, we give three new constructions of BH(G, h) matrices. The first construction is the first known family

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph Ks,t, we show that \({\rm{ex}}(n,K_{s,t}^\prime ) = O({n^{3/2 - {1 \over {2s}}}})\). This proves a conjecture of Kang, Kim

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph Ks,t, we show that \({\rm{ex}}(n,K_{s,t}^\prime) = O({n^{3/2 - {1 \over {2s}}}})\). This proves a conjecture of Kang, Kim