A note on long rainbow arithmetic progressions

Jungi\'{c} et al (2003) defined $T_{k}$ as the minimal number $t \in \mathbb{N}$ such that there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[t n]$ for every $n \in \mathbb{N}$. They proved that for every $k \geq 3$, $\lfloor \frac{k^2}{4} \rfloor < T_{k} \leq \frac{k(k-1)^2}{2}$ and conjectured that $T_{k} = \Theta(k^2)$. We prove for all $\epsilon > 0$ that $T_{k} = O(k^{5/2+\epsilon})$ using the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem and Wigert's bound on the divisor function.Comment: 3 page

Further results on discrete unitary invariance

In arXiv:1607.06679, Marcus proved that certain functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. In this note, we generalize the results from the Marcus paper to a larger class of functions of multiple matrices. We also answer a problem posed in the Marcus paper.Comment: 7 page

Improved lower bound on generalized Erdos-Ginzburg-Ziv constants

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then $s_{2n}(C_{n}^{r}) > \frac{n}{2}[\frac{5}{4}-O(n^{-\frac{3}{2}})]^{r}$ and $s_{k n}(C_{n}^{r}) > \frac{k n}{4} [1+\frac{1}{e k}-O(\frac{1}{n})]^{r}$ for $k > 2$. In this note, we sharpen their general bound by showing that $s_{k n}(C_{n}^{r}) > \frac{k n}{4} [1+\frac{(k-1)^{(k-1)}}{k^k}-O(\frac{1}{n})]^{r}$ for $k > 2$.Comment: 3 page

Asymptotic bounds on renewal process stopping times

Suppose that i.i.d. random variables $X_{1}, X_{2}, \ldots$ are chosen uniformly from $[0,1]$, and let $f: [0,1] \rightarrow [0,1]$ be an increasing bijection. Define $\mu_{f}$ to be the expected value of $f(X_{i})$ for each $i$. Define the random variable $K_{f}$ be to be minimal so that $\sum_{i = 1}^{K_{f}} f(X_{i}) > t$ and let $N_{f}(t)$ be the expected value of $K_{f}$. We prove that if $c_{f} = \frac{\int_{0}^{1} \int_{f^{-1}(u)}^{1} (f(x)-u) dx du}{\mu_{f}}$, then $N_{f}(t) = \frac{t+c_{f}}{\mu_{f}}+o(1)$. This generalizes a result of \'{C}urgus and Jewett (2007) on the case $f(x) = x$.Comment: 8 page

Constructing sparse Davenport-Schinzel sequences

For any sequence $u$, the extremal function $Ex(u, j, n)$ is the maximum possible length of a $j$-sparse sequence with $n$ distinct letters that avoids $u$. We prove that if $u$ is an alternating sequence $a b a b \dots$ of length $s$, then $Ex(u, j, n) = \Theta(s n^{2})$ for all $j \geq 2$ and $s \geq n$, answering a question of Wellman and Pettie [Lower Bounds on Davenport-Schinzel Sequences via Rectangular Zarankiewicz Matrices, Disc. Math. 341 (2018), 1987--1993] and extending the result of Roselle and Stanton that $Ex(u, 2, n) = \Theta(s n^2)$ for any alternation $u$ of length $s \geq n$ [Some properties of Davenport-Schinzel sequences, Acta Arithmetica 17 (1971), 355--362]. Wellman and Pettie also asked how large must $s(n)$ be for there to exist $n$-block $DS(n, s(n))$ sequences of length $\Omega(n^{2-o(1)})$. We answer this question by showing that the maximum possible length of an $n$-block $DS(n, s(n))$ sequence is $\Omega(n^{2-o(1)})$ if and only if $s(n) = \Omega(n^{1-o(1)})$. We also show related results for extremal functions of forbidden 0-1 matrices with any constant number of rows and extremal functions of forbidden sequences with any constant number of distinct letters

Bounds for approximating lower envelopes with polynomials of degree at most dd

Given a lower envelope in the form of an arbitrary sequence $u$, let $LSP(u, d)$ denote the maximum length of any subsequence of $u$ that can be realized as the lower envelope of a set of polynomials of degree at most $d$. Let $sp(m, d)$ denote the minimum value of $LSP(u, d)$ over all sequences $u$ of length $m$. We derive bounds on $sp(m, d)$ using another extremal function for sequences. A sequence $u$ is called $v$-free if no subsequence of $u$ is isomorphic to $v$. Given sequences $u$ and v, let $LSS(u, v)$ denote the maximum length of a $v$-free subsequence of $u$. Let $ss(m, v)$ denote the minimum of $LSS(u, v)$ over all sequences $u$ of length $m$. By bounding $ss(m, v)$ for alternating sequences $v$, we prove quasilinear bounds in $m^{1/2}$ on $sp(m,d)$ for all $d > 0$.Comment: 9 page

Improved lower bounds on extremal functions of multidimensional permutation matrices

A $d$-dimensional zero-one matrix $A$ avoids another $d$-dimensional zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeroes. Let $f(n,P,d)$ denote the maximum number of ones in a $d$-dimensional $n \times \cdots \times n$ zero-one matrix that avoids $P$. Fox proved for $n$ sufficiently large that $f(n, P, 2) = 2^{k^{\Theta(1)}}n$ for almost all $k \times k$ permutation matrices $P$. We extend this result by proving for $d \geq 2$ and $n$ sufficiently large that $f(n, P, d) = 2^{k^{\Theta(1)}}n^{d-1}$ for almost all $d$-dimensional permutation matrices $P$ of dimensions $k \times \cdots \times k$.Comment: 8 page

Metric dimension and pattern avoidance in graphs

In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$ is at most $(\lfloor \frac{2D}{3}\rfloor +1)^{k}+k \sum_{i = 1}^{\lceil \frac{D}{3}\rceil } (2i)^{k-1}$, sharpening the bound of $\binom{k}{2}+k D^{k-1}+D^{k}$ from Zubrilina (2018). We also show that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains the complete graph $K_{n}$ as a subgraph is $n = 2^{k}$. We prove that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains the complete bipartite graph $K_{n,n}$ as a subgraph is $2^{\Theta(k)}$. Furthermore, we show that the maximum value of $n$ for which some graph of edge metric dimension $\leq k$ contains $K_{1,n}$ as a subgraph is $n = 2^{k}$. We also show that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains $K_{1,n}$ as a subgraph is $3^{k}-O(k)$. In addition, we prove that the $d$-dimensional grids $\prod_{i = 1}^{d} P_{r_{i}}$ have edge metric dimension at most $d$. This generalizes two results of Kelenc et al. (2016), that non-path grids have edge metric dimension $2$ and that $d$-dimensional hypercubes have edge metric dimension at most $d$. We also provide a characterization of $n$-vertex graphs with edge metric dimension $n-2$, answering a question of Zubrilina. As a result of this characterization, we prove that any connected $n$-vertex graph $G$ such that $edim(G) = n-2$ has diameter at most $5$. More generally, we prove that any connected $n$-vertex graph with edge metric dimension $n-k$ has diameter at most $3k-1$

Improved bounds on maximum sets of letters in sequences with forbidden alternations

Let $A_{s,k}(m)$ be the maximum number of distinct letters in any sequence which can be partitioned into $m$ contiguous blocks of pairwise distinct letters, has at least $k$ occurrences of every letter, and has no subsequence forming an alternation of length $s$. Nivasch (2010) proved that $A_{5, 2d+1}(m) = \theta( m \alpha_{d}(m))$ for all fixed $d \geq 2$. We show that $A_{s+1, s}(m) = \binom{m- \lceil \frac{s}{2} \rceil}{\lfloor \frac{s}{2} \rfloor}$ for all $s \geq 2$, $A_{5, 6}(m) = \theta(m \log \log m)$, and $A_{5, 2d+2}(m) = \theta(m \alpha_{d}(m))$ for all fixed $d \geq 3$.Comment: 10 page

Bounding extremal functions of forbidden 0−10-1 matrices using (r,s)(r,s)-formations

First, we prove tight bounds of $n 2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ on the extremal function of the forbidden pair of ordered sequences $(1 2 3 \ldots k)^t$ and $(k \ldots 3 2 1)^t$ using bounds on a class of sequences called $(r,s)$-formations. Then, we show how an analogous method can be used to derive similar bounds on the extremal functions of forbidden pairs of $0-1$ matrices consisting of horizontal concatenations of identical identity matrices and their horizontal reflections.Comment: 10 page