Related papers: How many double squares can a string contain?

We show that the number of integer solutions for a pair of bilinear equations in at least 2*6 variables has (up to logarithms) the expected upper bound unless there is a structural reason why it is not the case.

The number of apparent double points of a smooth, irreducible projective variety $X$ of dimension $n$ in $\Proj^{2n+1}$ is the number of secant lines to $X$ passing through the general point of $\Proj^{2n+1}$. This classical notion dates…

Motivated by mass-spectrometry protein sequencing, we consider a simply-stated problem of reconstructing a string from the multiset of its substring compositions. We show that all strings of length 7, one less than a prime, or one less than…

We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2),... n(k),.... Extending and improving previous work of Nurmela…

Chung and Graham [J. London Math. Soc., 1983] claimed that there exists an $n$-vertex graph $G$ containing all $n$-vertex trees as subgraphs that has at most $\frac{5}{2}n \log_2 n + O(n)$ edges. We identify an error in their proof. This…

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…

We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree $\Delta \leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-, 12-,…

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$…

We study a problem of Santos about the largest possible diameter of a $d$-dimensional (abstract) simplicial complex on $n$ vertices. For dimension 2, we determine the exact value of the maximum for every $n$ using an explicit construction.…

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Kr\'al' and Lamaison [arXiv, September 2022], and an upper…

String graphs, that is, intersection graphs of curves in the plane, have been studied since the 1960s. We provide an expository presentation of several results, including very recent ones: some string graphs require an exponential number of…

We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for some $k$) instead of cycles (graphs with all degrees even). We give an…

In 1975, Erd\H{o}s asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erd\H{o}s observed that the upper bound $O(n^{5/3})$ holds since…

We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to…

Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao has shown recently that the maximum possible…

Let $n(k_1, k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that ${n(k,…

A graph {\it has cutwidth at most 2} if one can number its vertices by $1,\ldots n$ so that for every $i=1,\ldots,n-1$ there are at most 2 edges $(u,v)$ such that $u\le i<v$. A characterization of graphs having cutwidth at most 2 in terms…

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely…

The Sum of Squares algorithm for bin packing was defined in [2] and studied in great detail in [1], where it was proved that its worst case performance ratio is at most 3. In this note, we improve the asymptotic worst case bound to…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve…