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

We prove for every graph H there exists ɛ > 0 such that, for every graph G with |G|≥2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least ɛ|G| neighbours, or there are two disjoint sets A, B ⊆ V(G) with |A|,|B|≥ɛ|G| such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or

We present the solution of a long-standing open question by giving an explicit construction of an infinite family of \(\mathbb{M}\)-vertex cubic graphs that have diameter \(\left[ {1 + o\left( 1 \right)} \right]{\log _2}\mathbb{M}\). Then, for every K in the form K = ps + 1, where p can be any prime [including 2] and s any positive integer, we extend the techniques to construct an infinite family of

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given r-edge-coloured graph G? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question

We prove a quantitative, finitary version of Trofimov’s result that a connected, locally finite vertex-transitive graph Γ of polynomial growth admits a quotient with finite fibres on which the action of Aut(Γ) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph Γ of large diameter admits a quotient with fibres

We extend the notion of tournaments to orientation of the (k − 1)-skeleton of an (n − 1)-dimensional simplex. We ask for the maximal number of k-simplices whose boundary matches the orientation, extending the question on the upper bound of the number of directed 3-cycles of a tournament. In the general case we show polynomial upper and lower bounds. For the case k = 3 we improve the lower bound. Furthermore

In this paper, we complete the classification of transitive ovoids of finite Hermitian polar spaces.

We prove that for every integer t ⩾ 1 there exists a constant ct such that for every Kt-minor-free graph G, and every set S of balls in G, the minimum size of a set of vertices of G intersecting all the balls of S is at most ct times the maximum number of vertex-disjoint balls in S. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having

A well-known theorem of Chung and Graham states that if h ≥ 4 then a tournament T is quasirandom if and only if T contains each h-vertex tournament the ‘correct number’ of times as a subtournament. In this paper we investigate the relationship between quasirandomness of T and the count of a single h-vertex tournament H in T. We consider two types of counts, the global one and the local one. We first

In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than ωω. Big Ramsey degrees of finite chains in all other countable ordinals are infinite.

In this paper, we generalize the so-called Korchmáros—Mazzocca arcs, that is, point sets of size q + t intersecting each line in 0, 2 or t points in a finite projective plane of order q. For t ≠ 2, this means that each point of the point set is incident with exactly one line meeting the point set in t points. In PG(2, pn), we change 2 in the definition above to any integer m and describe all examples

For a vector space \({\mathbb{F}^n}\) over a field \(\mathbb{F}\), an (η, β)-dimension expander of degree d is a collection of d linear maps \({\Gamma _j}:{\mathbb{F}^n} \rightarrow {\mathbb{F}^n}\) such that for every subspace U of \({\mathbb{F}^n}\) of dimension at most ηn, the image of U under all the maps, ∑ dj=1 Γj(U), has dimension at least α dim(U). Over a finite field, a random collection of

An (n, d, λ)-graph is a d regular graph on n vertices in which the absolute value of any nontrivial eigenvalue is at most λ. For any constant d ≥ 3, ϵ > 0 and all sufficiently large n we show that there is a deterministic poly(n) time algorithm that outputs an (n, d, λ)-graph (on exactly n vertices) with \(\lambda \le 2\sqrt {d - 1} + \). For any d=p + 2 with p ≡ 1 mod 4 prime and all sufficiently

We prove a bijection between the triangulations of the 3-dimensional cyclic poly-tope C(n + 2, 3) and persistent graphs with n vertices. We show that under this bijection the Stasheff-Tamari orders on triangulations naturally translate to subgraph inclusion between persistent graphs. Moreover, we describe a connection to the second higher Bruhat order B(n, 2). We also give an algorithm to efficiently

Gallai asked in 1984 if any k-critical graph on n vertices contains at least n distinct (k − 1)-critical subgraphs. The answer is trivial for k ≤ 3. Improving a result of Stiebitz [10], Abbott and Zhou [1] proved in 1995 that for all k ≥ 4, any k-critical graph contains Ω(n1/(k − 1)) distinct (k − 1)-critical subgraphs. Since then no progress had been made until very recently, when Hare [4] resolved

Let V be an r-dimensional \({\mathbb{F}_{{q^n}}}\)-vector space. We call an \({\mathbb{F}_{q}}\)-subspace U of V h-scattered if U meets the h-dimensional \({\mathbb{F}_{{q^n}}}\)-subspaces of V in \({\mathbb{F}_{q}}\)-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that \({\dim_{\mathbb{F}_{q}}}\) U ≤ rn/2 when U is 1-scattered. Sub-spaces attaining this bound have been investigated

We construct a family of PL triangulations of the d-dimensional real projective space ℝPd on \(\Theta \left( {{{\left( {\frac{{1 + \sqrt 5 }}{2}} \right)}^{d + 1}}} \right)\) vertices for every d s-> 1. This improves a construction due to Kühnel on 2d+1 -1 vertices.

For a positive constant α a graph G on n vertices is called an α-expander if every vertex set U of size at most n/2 has an external neighborhood whose size is at least α|U|. We study cycle lengths in expanding graphs. We first prove that cycle lengths in α-expanders are well distributed. Specifically, we show that for every 0 < α ⪯ 1 there exist positive constants n0, C and A = O(1/α) such that for

In 1985, Ben-Or and Linial (Advances in Computing Research 1989) introduced the collective coin flipping problem, where n parties communicate via a single broadcast channel and wish to generate a common random bit in the presence of adaptive Byzantine corruptions. In this model, the adversary can decide to corrupt a party in the course of the protocol as a function of the messages seen so far. They

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $\delta(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y), \mathbb{P}(y\prec x)\}$, where $\mathbb{P}(x\prec y)$ is the probability $x$ is less than $y$ in a uniformly random linear extension of $P$. In particular, we show that if $P$

In this short note, we show that the VC-dimension of the class of $k$-vertex polytopes in $\mathbb R^d$ is at most $8d^2k\log_2k$, answering an old question of Long and Warmuth.

Robertson and Seymour proved that the family of all graphs containing a fixed graph H as a minor has the Erdős-Posa property if and only if H is planar. We show that this is no longer true for the edge version of the Erdős-Posa property, and indeed even fails when H is an arbitrary subcubic tree of large pathwidth or a long ladder. This answers a question of Raymond, Sau and Thilikos.

In his PhD thesis Shih characterized the relationship between two graphs, where the cycle space of the first is included in the cycle space of the second and the dimension of the cycle spaces differ by one [7]. However, this result never appeared in a refereed publication. As a consequence this theorem has not received the attention it deserves. We give a simpler and shorter proof of this result as

We prove the conjecture of Seymour (1993) that for every apex-forest $H_1$ and outerplanar graph $H_2$ there is an integer $p$ such that every 2-connected graph of pathwidth at least $p$ contains $H_1$ or $H_2$ as a minor. An independent proof was recently obtained by Dang and Thomas.

Let ${\mathcal{M} = (M_i \colon i\in K)}$ be a finite or infinite family consisting of matroids on a common ground set $E$ each of which may be finitary or cofinitary. We prove the following Cantor-Bernstein-type result: If there is a collection of bases, one for each $M_i$, which covers the set $E$, and also a collection of bases which is pairwise disjoint, then there is a collection of bases which

A $d$-simplex is defined to be a collection $A_1,\dots,A_{d+1}$ of subsets of size $k$ of $[n]$ such that the intersection of all of them is empty, but the intersection of any $d$ of them is non-empty. Furthemore, a $d$-cluster is a collection of $d+1$ such sets with empty intersection and union of size $\le 2k$, and a $d$-simplex-cluster is such a collection that is both a $d$-simplex and a $d$-cluster

Let p ∈ ℤ [x] be any polynomial with p(0) =0, k ∈ ℕ and let c1, …, cs ∈ ℤ, s ⩾ k(k + 1), be non-zero integers such that \(\sum {{c_i} = 0} \). We show that for a wide class of coefficients c1, …, cs in every finite coloring ℕ = A1∪ ⃯ ∪ Ar there is a monochromatic solution to the equation \({c_1}x_1^k + \cdots + {c_s}x_s^k = {\rm{p}}\left(y \right).\).

For each integer $t\ge 5$, we give a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least $t$ and odd.

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(n)

The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions

We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.

Let ℋ be the class of bounded measurable symmetric functions on [0, 1] 2 . For a function h ∈ ℋ and a graph G with vertex set [ v 1 ,⌦, v n } and edge set E ( G ), define $${t_G}(h) = \int \cdots \int {\prod\limits_{{\rm{\{ }}{v_i}{\rm{,}}{v_j}{\rm{\} }} \in E(G)} {h({x_i},{x_j})\;d{x_1} \cdots d{x_n}} } .$$ t G ( h ) = ∫ ⋯ ∫ ∏ { v i , v j } ∈ E ( G ) h ( x i , x j ) d x 1 ⋯ d x n . Answering a question

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if G is such a graph, then G 2 is (Δ( G ) + 72)-degenerate. This implies an upper bound of Δ( G ) ∣ 73 on the chromatic number of G 2 as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon-Tarsi number, and correspondence chromatic number. We also

Dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if P has dimension d, then to know whether x ≤ y in P it is enough to check whether x ≤ y in each of the d linear extensions of a witnessing realizer. Focusing on the encoding aspect, Nešetřil and Pudlák defined a more expressive version of dimension. A poset P has Boolean dimension

We prove the following statement. Let f ∈ ℝ[ x 1 ,…, x d ], for some d ≥ 3, and assume that f depends non-trivially in each of x 1 ,…, x d . Then one of the following holds. (i) For every finite sets A 1 ,…, A d ⊂ℝ, each of size n , we have $$\left| {f\left( {{A_1} \times \ldots \times {A_d}} \right)} \right| = \Omega \left( {{n^{3/2}}} \right),$$ | f ( A 1 × … × A d ) | = Ω ( n 3 / 2 ) , with constant