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

Colour the edges of the complete graph with vertex set \({\{1, 2, \dotsc , n\}}\) with an arbitrary number of colours. What is the smallest integer f(l, k) such that if \(n > f(l,k)\) then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that \(f(l, k) = l^{k - 1}\) and proved this for \(l \ge (3 k)^{2 k}\)

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition

In this paper we prove that every totally real algebraic integer \(\lambda \) of degree \(d \ge 2\) occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\). In order to prove this, for a given algebraic number \(\alpha \ne 0\), we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) for \(x \ne 1\). The problem of finding the smallest

Given a (thick) irreducible spherical building \(\Omega \), we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long root geometry

Let I(G) be the edge ideal of a simple graph G. We prove that \(I(G)^2\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 3\). Similarly, we show that \(I(G)^3\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 4\). We deduce these characterizations by establishing a general formula for the regularity of powers

Fuglede’s conjecture states that a subset \(\Omega \subseteq \mathbb {R}^{n}\) with positive and finite Lebesgue measure is a spectral set if and only if it tiles \(\mathbb {R}^{n}\) by translation. However, this conjecture does not hold in both directions for \(\mathbb {R}^n\), \(n\ge 3\). While the conjecture remains unsolved in \(\mathbb {R}\) and \(\mathbb {R}^2\), cyclic groups are instrumental

Let A be a subset of the cyclic group \({\textbf{Z}}/p{\textbf{Z}}\) with p prime. It is a well-studied problem to determine how small |A| can be if there is no unique sum in \(A+A\), meaning that for every two elements \(a_1,a_2\in A\), there exist \(a_1',a_2'\in A\) such that \(a_1+a_2=a_1'+a_2'\) and \(\{a_1,a_2\}\ne \{a_1',a_2'\}\). Let m(p) be the size of a smallest subset of \({\textbf{Z}}/p{\textbf{Z}}\)

Let \(n \ge 2k \ge 4\) be integers, \({[n]\atopwithdelims ()k}\) the collection of k-subsets of \([n] = \{1, \ldots , n\}\). Two families \({\mathcal {F}}, {\mathcal {G}} \subset {[n]\atopwithdelims ()k}\) are said to be cross-intersecting if \(F \cap G \ne \emptyset \) for all \(F \in {\mathcal {F}}\) and \(G \in {\mathcal {G}}\). A family is called non-trivial if the intersection of all its members