Given an n-element set \(C\subseteq \mathbb {R}^d\) and a (sufficiently generic) k-element multiset \(V\subseteq \mathbb {R}^d\), we can order the points in C by ranking each point \(c\in C\) according to the sum of the distances from c to the points of V. Let \(\Psi _k(C)\) denote the set of orderings of C that can be obtained in this manner as V varies, and let \(\psi ^{\textrm{max}}_{d,k}(n)\) be

A subset S of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of S are distinct. Imre Ruzsa asked the following question: What is the maximum number f(N) such that every set S of N real numbers contains a bi-Sidon subset of size at least f(N)? He proved that \(f(N)\geqslant cN^{\frac{1}{3}}\)

Jordán and Tanigawa recently introduced the d-dimensional algebraic connectivity \(a_d(G)\) of a graph G. This is a quantitative measure of the d-dimensional rigidity of G which generalizes the well-studied notion of spectral expansion of graphs. We present a new lower bound for \(a_d(G)\) defined in terms of the spectral expansion of certain subgraphs of G associated with a partition of its vertices

In this paper, we investigate the hypergraph Turán number \(\textrm{ex}(n,K^{(r)}_{s,t})\). Here, \(K^{(r)}_{s,t}\) denotes the r-uniform hypergraph with vertex set \(\left( \cup _{i\in [t]}X_i\right) \cup Y\) and edge set \(\{X_i\cup \{y\}: i\in [t], y\in Y\}\), where \(X_1,X_2,\cdots ,X_t\) are t pairwise disjoint sets of size \(r-1\) and Y is a set of size s disjoint from each \(X_i\). This study

In the max–min allocation problem a set P of players are to be allocated disjoint subsets of a set R of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezáková and Dani (SIGecom Exch 5(3):11–18

For two graphs F, H and a positive integer n, the function \(f_{F,H}(n)\) denotes the largest m such that every H-free graph on n vertices contains an F-free induced subgraph on m vertices. This function has been extensively studied in the last 60 years when F and H are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic

In 2002, D. Fon-Der-Flaass constructed a prolific family of strongly regular graphs. In this paper, we prove that for infinitely many natural numbers n and a positive constant c, this family contains at least \(n^{c\cdot n^{2/3}}\) strongly regular n-vertex graphs X with the same parameters, which satisfy the following condition: an isomorphism between X and any other graph can be verified by the 4-dimensional

We generalize the (signed) Varchenko matrix of a hyperplane arrangement to complexes of oriented matroids and show that its determinant has a nice factorization. This extends previous results on hyperplane arrangements and oriented matroids.

We prove that in every metric space where no line contains all the points, there are at least \(\Omega (n^{2/3})\) lines. This improves the previous \(\Omega (\sqrt{n})\) lower bound on the number of lines in general metric space, and also improves the previous \(\Omega (n^{4/7})\) lower bound on the number of lines in metric spaces generated by connected graphs.

Nešetřil and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field

An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping \(f:E(G) \rightarrow E(H)\) such that each r adjacent edges of G are mapped to r adjacent edges of H. For every \(r\ge 3\), let \(\mathcal H_r\) be an inclusion-wise minimal set of connected r-graphs, such that for every connected

As a first step towards a conjecture of Kahle and Newman, we prove that if \(T_n\) is a random 2-dimensional determinantal hypertree on n vertices, then $$\begin{aligned} \frac{\dim H_1(T_n,\mathbb {F}_2)}{n^2} \end{aligned}$$ converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the 1-out 2-complex. Our proof relies on the large deviation

The supersaturation problem for a given graph F asks for the minimum number \(h_F(n,q)\) of copies of F in an n-vertex graph with \(\textrm{ex}(n,F)+q\) edges. Subsequent works by Rademacher, Erdős, and Lovász and Simonovits determine the optimal range of q (which is linear in n) for cliques F such that \(h_F(n,q)\) equals the minimum number \(t_F(n,q)\) of copies of F obtained from a maximum F-free

Given integers \(n> k > 0\), and a set of integers \(L \subset [0, k-1]\), an L-system is a family of sets \(\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) such that \(|F \cap F'| \in L\) for distinct \(F, F'\in \mathcal {F}\). L-systems correspond to independent sets in a certain generalized Johnson graph G(n, k, L), so that the maximum size of an L-system is equivalent

We prove that, for any \(B \subset {\mathbb {R}}\), the Cartesian product set \(B \times B\) determines \(\Omega (|B|^{2+c})\) distinct angles.

Let G be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the \(S_4\)-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of G such that the complement of their union is a bipartite subgraph of G. They actually show that given any \(1^+\)-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge

Given a residually connected incidence geometry \(\Gamma \) that satisfies two conditions, denoted \((B_1)\) and \((B_2)\), we construct a new geometry \(H(\Gamma )\) with properties similar to those of \(\Gamma \). This new geometry \(H(\Gamma )\) is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how \(H(\Gamma )\) relates

For an odd integer \(n = 2d-1\), let \({\mathcal {B}}_d\) be the subgraph of the hypercube \(Q_n\) induced by the two largest layers. In this paper, we describe the typical structure of proper q-colorings of \(V({\mathcal {B}}_d)\) and give asymptotics on the number of such colorings when q is an even number. The proofs use various tools including information theory (entropy), Sapozhenko’s graph container

A spherical L-code, where \(L \subseteq [-1,\infty )\), consists of unit vectors in \(\mathbb {R}^d\) whose pairwise inner products are contained in L. Determining the maximum cardinality \(N_L(d)\) of an L-code in \(\mathbb {R}^d\) is a fundamental question in discrete geometry and has been extensively investigated for various choices of L. Our understanding in high dimensions is generally quite poor

A permutation \(\pi \in \mathbb {S}_n\) is k-balanced if every permutation of order k occurs in \(\pi \) equally often, through order-isomorphism. In this paper, we explicitly construct k-balanced permutations for \(k \le 3\), and every n that satisfies the necessary divisibility conditions. In contrast, we prove that for \(k \ge 4\), no such permutations exist. In fact, we show that in the case \(k

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation

We determine the excluded minors characterising the class of countable graphs that embed into some compact surface.

In this paper, we prove that the Fourier entropy of an n-dimensional boolean function f can be upper-bounded by \(O(I(f)+ \sum \limits _{k\in [n]}I_k(f)\log \frac{1}{I_k(f)})\), where I(f) is its total influence and \(I_k(f)\) is the influence of the k-th coordinate. There is no strict quantitative relationship between our bound with the known bounds for the Fourier-Min-Entropy-Influence conjecture

A subset \(\mathcal {C}\subseteq \{0,1,2\}^n\) is said to be a trifferent code (of block length n) if for every three distinct codewords \(x,y, z \in \mathcal {C}\), there is a coordinate \(i\in \{1,2,\ldots ,n\}\) where they all differ, that is, \(\{x(i),y(i),z(i)\}\) is same as \(\{0,1,2\}\). Let T(n) denote the size of the largest trifferent code of block length n. Understanding the asymptotic behavior

A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex v and every color \(\alpha \), there are at most as many edges incident to v colored with \(\alpha \) as with all other colors. We extend some known results for finite graphs to infinite graphs, also in the list setting. In particular, we prove that every infinite graph without pendant edges

In this paper we give anticoncentration bounds for sums of independent random vectors in finite-dimensional vector spaces. In particular, we asymptotically establish a conjecture of Leader and Radcliffe (SIAM J Discrete Math 7:90–101, 1994) and a question of Jones (SIAM J Appl Math 34:1–6, 1978). The highlight of this work is an application of the strong perfect graph theorem by Chudnovsky et al. (Ann

For all integers \(n \ge k > d \ge 1\), let \(m_{d}(k,n)\) be the minimum integer \(D \ge 0\) such that every k-uniform n-vertex hypergraph \({\mathcal {H}}\) with minimum d-degree \(\delta _{d}({\mathcal {H}})\) at least D has an optimal matching. For every fixed integer \(k \ge 3\), we show that for \(n \in k \mathbb {N}\) and \(p = \Omega (n^{-k+1} \log n)\), if \({\mathcal {H}}\) is an n-vertex

A storage code on a graph G is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate

We show that, for any prime p and integer \(k \ge 2\), a simple \({{\,\textrm{GF}\,}}(p)\)-representable matroid with sufficiently high rank has a rank-k flat which is either independent in M, or is a projective or affine geometry. As a corollary we obtain a Ramsey-type theorem for \({{\,\textrm{GF}\,}}(p)\)-representable matroids. For any prime p and integer \(k\ge 2\), if we 2-colour the elements

Given a graph G with a set F(v) of forbidden values at each \(v \in V(G)\), an F-avoiding orientation of G is an orientation in which \(\deg ^+(v) \not \in F(v)\) for each vertex v. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if \(|F(v)| < \frac{1}{2} \deg (v)\) for each \(v \in V(G)\), then G has an F-avoiding orientation, and they showed that this statement is true when \(\frac{1}{2}\)

A family \(\mathcal {F} \subset \mathcal {P}(n)\) is r-wise k-intersecting if \(|A_1 \cap \dots \cap A_r| \ge k\) for any \(A_1, \dots , A_r \in \mathcal {F}\). It is easily seen that if \(\mathcal {F}\) is r-wise k-intersecting for \(r \ge 2\), \(k \ge 1\) then \(|\mathcal {F}| \le 2^{n-1}\). The problem of determining the maximum size of a family \(\mathcal {F}\) that is both \(r_1\)-wise \(k_1\)-intersecting

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph \(K_{k,k}\) as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is \(O\left( n^{2 - 1/k}\right) \). An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most

We prove that every properly edge-colored n-vertex graph with average degree at least \(32(\log 5n)^2\) contains a rainbow cycle, improving upon the \((\log n)^{2+o(1)}\) bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least \(10^5 k^3 n^{1+1/k}\) edges contains a rainbow 2k-cycle, which improves the previous bound \(2^{ck^2}n^{1+1/k}\) obtained by Janzer.

We prove that every additive set A with energy \(E(A)\ge |A|^3/K\) has a subset \(A'\subseteq A\) of size \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\) such that \(|A'-A'|\le O_\varepsilon (K^{4}|A'|)\). This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc uv in a tournament T is the set of vertices that form a directed triangle with arc uv. We show that if the neighborhood of

When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded “tree-length”. We will show that this is equivalent to being “boundedly quasi-isometric to a tree”, which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to