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

How many squares are spanned by $n$ points in the plane? Here we study the corresponding maximum possible number $S_{\square}(n)$ of squares and determine the exact values for all $n\le 17$. For $18\le n\le 100$ we give lower bounds for…

There are at present two known string theories in $(2,2)$ dimensions. One of them is the well known $N=2$ string, and the other one is a more recently constructed $N=1$ spacetime supersymmetric string. They are both based on certain…

This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$ are integers, can be tiled (exactly packed) by squares where each has an integer side length of at least 2. In particular, we prove that tiling is always possible…

The combinatorics of squares in a word depends on how the equivalence of halves of the square is defined. We consider Abelian squares, parameterized squares, and order-preserving squares. The word $uv$ is an Abelian (parameterized,…

A finite word $w$ is an abelian square if $w = xx^\prime$ with $x^\prime$ a permutation of $x$. In 1972, Entringer, Jackson, and Schatz proved that every binary word of length $k^2 + 6k$ contains an abelian square of length $\geq 2k$. We…

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain $\Theta(n^2)$ distinct factors that are abelian squares. We study infinite words such that the number of abelian square…

The structural parameter twin-width was introduced by Bonnet et al. in [FOCS 2020], and already this first paper included an asymptotic argument bounding the twin-width of planar graphs by a non-explicit constant. Quite recently, we have…

We prove that the number of tangencies between the members of two families, each of which consists of $n$ pairwise disjoint curves, can be as large as $\Omega(n^{4/3})$. We show that from a conjecture about forbidden $0$-$1$ matrices it…

An order-preserving square in a string is a fragment of the form $uv$ where $u\neq v$ and $u$ is order-isomorphic to $v$. We show that a string $w$ of length $n$ over an alphabet of size $\sigma$ contains $\mathcal{O}(\sigma n)$…

We show a $n^2 \cdot 2^{n/2}$ upper bound on the number of $(132,213)$ avoiding cyclic permutations. This is the first nontrivial upper bound on the number of such permutations. We also construct an algorithm to determine whether a…

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+32t-1$$ for…

A {\em thrackle} is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of $n$ vertices has at most $1.3984n$ edges. {\em…

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate…

A homothetic packing of squares is any set of various-size squares with the same orientation where no two squares have overlapping interiors. If all $n$ squares have the same size then we can have up to roughly $4n$ contacts by arranging…

Put n nonoverlapping squares inside the unit square. Let f(n) and g(n) denote the maximum values of the sum of the edge lengths of the n small squares, where in the case of f(n) the maximum is taken over all arbitrary packings of the unit…

A square is a concatenation of two identical words, and a word $w$ is said to have a square $yy$ if $w$ can be written as $xyyz$ for some words $x$ and $z$. It is known that the ratio of the number of distinct squares in a word to its…

The online square detection problem is to detect the first occurrence of a square in a string whose characters are provided as input one at a time. Recall that a square is a string that is a concatenation of two identical strings. In this…

We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| >= 4; our construction is somewhat simpler than the original…

A circular word, or a necklace, is an equivalence class under conjugation of a word. A fundamental question concerning regularities in standard words is bounding the number of distinct squares in a word of length $n$. The famous conjecture…

Squares (fragments of the form $xx$, for some string $x$) are arguably the most natural type of repetition in strings. The basic algorithmic question concerning squares is to check if a given string of length $n$ is square-free, that is,…