Related papers: Constructing sparse Davenport-Schinzel sequences

Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure…

Given an alphabet $S$, we consider the size of the subsets of the full sequence space $S^{\rm {\bf Z}}$ determined by the additional restriction that $x_i\not=x_{i+f(n)},\ i\in {\rm {\bf Z}},\ n\in {\rm {\bf N}}.$ Here $f$ is a positive,…

Alon and Shikhelman initiated the systematic study of a generalization of the extremal function. Motivated by algorithmic applications, the study of the extremal function $\text{ex}(n, K_k, K_t\text{-minor})$, i.e., the number of cliques of…

Given a positive integer $n$, consider a random permutation $\tau$ of the set $\{1,2,\ldots, n\}$. In $\tau$, we look for sequences of consecutive integers that appear in adjacent positions: a maximal such a sequence is called a block. Each…

A border of a string is a non-empty prefix of the string that is also a suffix of the string, and a string is unbordered if it has no border other than itself. Loptev, Kucherov, and Starikovskaya [CPM 2015] conjectured the following: If we…

It is well-known that the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one…

For every integer $k \ge 3$, we determine the extremal structure of an $n$-vertex graph with at most $t$ vertex-disjoint copies of $C_{2k}$ when $n$ is sufficiently large and $t$ lies in the interval…

Let $\eta=(\eta(t))_{t\in T}$ be a sample continuous max-infinitely random field on a locally compact metric space $T$. For a closed subset $S\in T$, we note $\eta_{S}$ the restriction of $\eta$ to $S$. We consider $\beta(S_1,S_2)$ the…

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for…

The Damerau-Levenshtein distance between two sequences is the minimum number of operations (deletions, insertions, substitutions, and adjacent transpositions) required to convert one sequence into another. Notwithstanding a long history of…

For integers $n\ge s\ge2$, let $e(n,s)$ denote the maximum size of a family $\F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erd\H{o}s--Kleitman problem, is the non-uniform…

Defant and Zheng introduced a consecutive-pattern-avoiding stack sort map $SC_{\sigma}$, where the stack must avoid a consecutive pattern $\sigma$. Seidel and Sun disproved a conjecture in Defant and Zheng's paper about the maximum…

The Tur\'an number of a graph $F$, $ex(n,F)$, is the maximum number of edges in a graph on $n$ vertices which does not contain $F$ as a subgraph. Let $S_{a,b}$ denote a double star with a central edge $uv$, $a$ leaves connected to $u$ and…

The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) =…

Let $\beta > 1$ and the run-length function $r_n(x,\beta)$ be the maximal length of consecutive zeros amongst the first n digits in the $\beta$-expansion of $x\in[0,1]$. The exceptional set $$E_{\max}^{\varphi}=\left\{x \in…

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge…

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected…

In this paper, we provide an efficient algorithm to construct almost optimal $(k,n,d)$-superimposed codes with runlength constraints. A $(k,n,d)$-superimposed code of length $t$ is a $t \times n$ binary matrix such that any two 1's in each…

For positive integers $n$ and $m$, consider a multiset of non-empty subsets of $[m]$ such that there is a \textit{unique} partition of these subsets into $n$ partitions of $[m]$. We study the maximum possible size $g(n,m)$ of such a…

Let $K^{(r)}_{s_1,s_2,\cdots,s_r}$ be the complete $r$-partite $r$-uniform hypergraph and $ex(n,K^{(r)}_{s_1,s_2,\cdots,s_r})$ be the maximum number of edges in any $n$-vertex $K^{(r)}_{s_1,s_2,\cdots,s_r}$-free $r$-uniform hypergraph. It…