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

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 δ(P) = max(x,y)∈P2 min{ℙ(x ≺ y), ℙ(y ≺ x)}, where ℙ(x≺y) is the probability x is less than y in a uniformly random linear extension of P. In particular, we show that if P is a width 2 poset that cannot be formed from the singleton poset and

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 this short note, we show that the VC-dimension of the class of k-vertex polytopes in Rd is at most 8d2k log2k, answering an old question of Long and Warmuth. We also show that it is at least 1/3kd.

Robertson and Seymour proved that the family of all graphs containing a fixed graph H as a minor has the Erdős-Pósa property if and only if H is planar. We show that this is no longer true for the edge version of the Erdős-Pósa 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 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

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 a conjecture of Seymour (1993) stating that for every apex-forest H1 and out-erplanar graph H2 there is an integer p such that every 2-connected graph of pathwidth at least p contains H1 or H2 as a minor. An independent proof was recently obtained by Dang and Thomas [3].

Let M = (Mi: i ∈ 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 Mi, which covers the set E, and also a collection of bases which are pairwise disjoint, then there is a collection of bases which partition E. We also show

A d-simplex is defined to be a collection A1,..., Ad+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 ≤ 2k, and a d-simplex-cluster is such a collection that is both a d-simplex and a d-cluster. The Erdös-Chvátal d-simplex

for each integer ℓ ≥ 5, we give a polynomial-time algorithm to test whether a graph contains an induced cycle with length at least ℓ and odd.

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 \({c_1}x_1^k + \cdots + {c_s}x_s^k = {\rm{p}}\left(y \right).\).

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 multi-normal of order \(1/\sqrt

The Hall ratio of a graph G is the maximum of |V(H)|/α(H) over all subgraphs H of G. It is easy to see that 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

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 [v1,⌦,vn} and edge set E(G), define$${t_G}(h) = \int \cdots \int {\prod\limits_{{\rm{\{ }}{v_i}{\rm{,}}{v_j}{\rm{\} }} \in E(G)} {h({x_i},{x_j})\;d{x_1} \cdots d{x_n}} } .$$Answering a question raised by Conlon and Lee, we prove that in order for tG(∣h∣)1/∣E(G)∣ to be a norm on

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if G is such a graph, then G2 is (Δ(G) + 72)-degenerate. This implies an upper bound of Δ(G) ∣ 73 on the chromatic number of G2 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 show

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[x1,…,xd], for some d ≥ 3, and assume that f depends non-trivially in each of x1,…, xd. Then one of the following holds. (i) For every finite sets A1,…, Ad ⊂ℝ, each of size n, we have $$\left| {f\left( {{A_1} \times \ldots \times {A_d}} \right)} \right| = \Omega \left( {{n^{3/2}}} \right),$$ with constant of proportionality that depends on deg f. (ii) f is

Let G = (V, E) be a finite graph. For v ∈ V we denote by Gv the subgraph of G that is induced by v’s neighbor set. We say that G is (a,b)-regular for a>b> 0 integers, if G is a-regular and Gv is b-regular for every v ∈ V. Recent advances in PCP theory call for the construction of infinitely many (a,b)-regular expander graphs G that are expanders also locally. Namely, all the graphs {Gv ∣ v ∈ V} should

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫n2 log−3n, where a hinge is identified by fixing two pairwise distances in a point triple. This is achieved via strengthening (modulo a logn factor) of the Guth- Katz estimate for the number of pairwise intersections of lines in ℝ3, arising in the context of the plane Erdős distinct distance

We show that a quasirandom k-uniform hypergraph G has a tight Euler tour subject to the necessary condition that k divides all vertex degrees. The case when G is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the k-subsets of an n-set.

In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in ℝn, such that all the vertices of \(\overrightarrow 0\) are covered at least k times, and \({\left\{{0,1} \right\}^n}\backslash \left\{{\overrightarrow 0} \right\}\) is uncovered? The k = 1 case is the well-known Alon-Füredi theorem which says a minimum of n affine hyperplanes is required, which follows

Given a set of n points in ℝd, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the ℓp-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d = ω(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [6], Williams [48]

The Caccetta-Häggkvist conjecture implies that for every integer k ≥ 1, if G is a bipartite digraph, with n vertices in each part, and every vertex has out-degree more than n/(k+1), then G has a directed cycle of length at most 2k. If true this is best possible, and we prove this for k = 1, 2, 3, 4, 6 and all k ≥ 224,539.More generally, we conjecture that for every integer k ≥ 1, and every pair of

A famous theorem of Kirkman says that there exists a Steiner triple system of order n if and only if n ≡ 1,3 mod 6. In 1973, Erdős conjectured that one can find so-called ‘sparse’ Steiner triple systems. Roughly speaking, the aim is to have at most j−3 triples on every set of j points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth

Let X = (VX, EX) be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and EX is equipped with an involution which inverts the orientation. Each oriented edge is labelled by an element of a finite alphabet Σ. The labelling is assumed to be deterministic: edges with the same initial (resp. terminal) vertex have distinct labels. Furthermore

A clutter is intersecting if the members do not have a common element yet every two members intersect. It has been conjectured that for clutters without an intersecting minor, total primal integrality and total dual integrality of the corresponding set covering linear system must be equivalent. In this paper, we provide a polynomial characterization of clutters without an intersecting minor. One important

We show that any graph that is generically globally rigid in ℝd has a realization in ℝd that is both generic and universally rigid. This also implies that the graph also must have a realization in ℝd that is both infinitesimally rigid and universally rigid; such a realization serves as a certificate of generic global rigidity.Our approach involves an algorithm by Lovász, Saks and Schrijver that, for

A construction of Alon and Krivelevich gives highly pseudorandom Kk-free graphs on n vertices with edge density equal to Θ(n−1=(k−2)). In this short note we improve their result by constructing an infinite family of highly pseudorandom Kk-free graphs with a higher edge density of Θ(n−1=(k−1)).

We say that two graphs on the same vertex set are G-creating if their union (the union of their edges) contains G as a subgraph. Let Hn(G) be the maximum number of pairwise G-creating Hamiltonian paths of Kn. Cohen, Fachini and Körner proved$${n^{\frac{1}{2}n - o\left( n \right)}} \le {H_n}\left( {{C_4}} \right) \le {n^{\frac{3}{4}n + o\left( n \right)}}.$$In this paper we close the superexponential

Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. A standard tool to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this paper, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of

We prove that there is a constant C ≤ 6.614 such that every Boolean function of degree at most d (as a polynomial over ℝ) is a C·2d-junta, i.e., it depends on at most C·2d variables. This improves the d·2d-1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)].The bound of C·2d is tight up to the constant C, since a read-once decision tree of depth d depends on all 2d - 1 variables

Let A be a random m × n matrix over the finite field \(\mathbb{F}_q\) with precisely k non-zero entries per row and let \(y\in\mathbb{F}_q^m\) be a random vector chosen independently of A. We identify the threshold m/n up to which the linear system Ax = y has a solution with high probability and analyse the geometry of the set of solutions. In the special case q = 2, known as the random k-XORSAT problem

We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high order walks is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs.In this work we go beyond spectral

We prove a lower bound of Ω(n2/log2n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x1,...,xn). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([34]), who proved a lower bound of Ω(n4/3/log2n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized

We complete the classification of tight 4-designs in Hamming association schemes H(n,q), i.e., that of tight orthogonal arrays of strength 4, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight 4-design in H(n,q) and analyze its triple intersection numbers to conclude the non-existence in all open cases.

Let G be a graph, and \(v \in V(G)\) and \(S \subseteq V(G)\setminus{v}\) of size at least k. An important result on graph connectivity due to Perfect states that, if v and S are k-linked, then a (k−1)-link between a vertex v and S can be extended to a k-link between v and S such that the endvertices of the (k−1)-link are also the endvertices of the k-link. We begin by proving a generalization of Perfect's

A result due to Gyárfás, Hubenko, and Solymosi (answering a question of Erdős) states that if a graph G on n vertices does not contain K2,2 as an induced subgraph yet has at least \(c\left(\begin{array}{c}n\\ 2\end{array}\right)\) edges, then G has a complete subgraph on at least \(\frac{c^2}{10}n\) vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced K2,2

A 1-factor in an n-vertex graph G is a collection of \(\frac{n}{2}\) vertex-disjoint edges and a 1-factorization of G is a partition of its edges into edge-disjoint 1-factors. Clearly, a 1-factorization of G cannot exist unless n is even and G is regular (that is, all vertices are of the same degree). The problem of finding 1-factorizations in graphs goes back to a paper of Kirkman in 1847 and has

In 1995 Kim famously proved the Ramsey bound R(3, t) ≤ ct2/logt by constructing an n-vertex graph that is triangle-free and has independence number at most \(C\,\sqrt {n\,\log \,n} \). We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph Kn into a packing of such nearly optimal Ramsey R(3,t) graphs.More precisely

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph F we define the homomorphism threshold as the infimum over all α ∈ [0,1] such that every n-vertex F-free graph G with minimum degree at least αn has a homomorphic image H of bounded order (i.e. independent of n), which is F-free

In this paper, we extend and refine previous Turán-type results on graphs with a given circumference. Let Wn,k,c be the graph obtained from a clique Kc−k+1 by adding n − (c − k + 1) isolated vertices each joined to the same k vertices of the clique, and let f(n,k,c) = e(Wn,k,c). Improving a celebrated theorem of Erdős and Gallai [8], Kopylov [18] proved that for c \max \left\{ {f(n,3,c),f\left( {n

The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least cH log d/log log d, thus nearly confirming one and

We investigate infinite highly-arc-transitive digraphs with two additional properties, property Z and descendant-homogeneity. We show that if D is a highly-arc-transitive descendant-homogeneous digraph with property Z and F is the subdigraph spanned by the descendant sets of a line in D, then F is a locally finite 2-ended digraph with property Z. If, moreover, D has prime out-valency, then there is

It follows from a theorem of Lovász that if D is a finite digraph with r ∈ V(D), then there is a spanning subdigraph E of D such that for every vertex v ≠ r the following quantities are equal: the local connectivity from r to v in D, the local connectivity from r to v in E and the indegree of v in E.In infinite combinatorics cardinality is often an overly rough measure to obtain deep results and it