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

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 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

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

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

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 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

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

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 clustered chromatic number of a class of graphs is the minimum integer k such that for some integer c every graph in the class is k-colourable with monochromatic components of size at most c. We prove that for every graph H, the clustered chromatic number of the class of H-minor-free graphs is tied to the tree-depth of H. In particular, if H is connected with tree-depth t, then every H-minor-free

Let F ⊂ 2 [n] be a family in which any three sets have non-empty intersection and any two sets have at least 32 elements in common. The nearly best possible bound F ≤ 2n−2 is proved. We believe that 32 can be replaced by 3 and provide a simple-looking conjecture that would imply this.

A natural number is a binary k’ th power if its binary representation consists of k consecutive identical blocks. We prove, using tools from combinatorics, linear algebra, and number theory, an analogue of Waring’s theorem for sums of binary k’th powers. More precisely, we show that for each integer k> 2, there exists an effectively computable natural number n such that every sufficiently large multiple