arXiv subjects
Publications and source records attributed to Dylan King.
We give a nibble-free construction proving $R(3,k)\ge(1/200+o(1))k^2/\log k$. We also include Shearer's proof bounding the independence number of a triangle-free graph, which implies $R(3,k)\le (1+o(1))(k^2/\log k)$.
Consider the following online filling game. An array of length $n$ is initially empty. At each time step one observes an independent sample from $\mathrm{Unif}[0,1]$ and must either discard it or place it irrevocably into an empty position of the array, while preserving the constraint that the occupied entries are non-decreasing from left to right. Among all possible strategies, what is the optimal expected time required to fill the array? Let $v_n$ denote this optimal expected completion time. Our main result determines $v_n$ up to lower-order terms: \[ v_n=\left(\frac12+o(1)\right)n\log n. \] More precisely, no strategy, even if randomized and adaptive, can have expected completion time below $\left(\frac12-o(1)\right)n\log n$, while we provide an explicit deterministic strategy whose expected completion time is at most $\left(\frac12+o(1)\right)n\log n$. For comparison, the natural coupon-collector strategy, which partitions $[0,1]$ into $n$ equal intervals and reserves one array position for each interval, has expected completion time $(1+o(1))n\log n$. We also consider a with-replacement version of the game, in which previously placed entries may be overwritten. For this variant, we give a deterministic strategy with expected completion time $O(n\sqrt{\log n})$, thereby establishing a separation between the two models.
We present a flexible random construction which, for certain graphs $H$, is able to produce $H$-free graphs with edge density strictly larger than that of the $H$-free process, while simultaneously preserving pseudorandom properties and allowing a much easier analysis. As our main application, we use this construction to show that the off-diagonal Ramsey numbers satisfy $R(3,k)\ge \left(\frac12+o(1)\right)\frac{k^2}{\log{k}}$, improving the previously best bound $R(3,k)\ge \left(\frac13+o(1)\right)\frac{k^2}{\log{k}}$. While the best known upper bound is $R(3,k)\le \left(1+o(1)\right)\frac{k^2}{\log{k}}$, the constant of $\frac12$ has been conjectured to be asymptotically tight by multiple groups.
For an ordered graph $F$, denote the Tur\'an density by $\vec{\pi}(F)$. The relative Tur\'an density, denoted by $\rho(F)$, is the supremum over $\alpha \in [0,1]$ such that every ordered graph $G$ contains an $F$-free subgraph $G'$ with $e(G') \geq \alpha e(G)$. Reiher, R\"odl, Sales and Schacht showed that $\rho(P) = \vec{\pi}(P)/2$ and $\rho(K) = \vec{\pi}(K)$ for any ascending path $P$ or clique $K$. They asked if there are any ordered graphs $F$ with $\vec{\pi}(F)/2 < \rho(F) < \vec{\pi}(F)$. We answer this question in the affirmative by describing a family of such $F$. We also show that the relative Tur\'an densities of a large family of ordered matchings (including $\{\{1,6\}, \{2,3\}, \{4,5\}\}$ and $\{\{1,3\}, \{2,5\}, \{4,6\}\}$) are $0$.
Given a family of $3$-graphs $\mathcal{F}$, the uniform Tur\'{a}n density $\pi_{\therefore}(\mathcal{F})$ is defined as the infimum $d\in[0,1]$ for which any sufficiently large uniformly $d$-dense $3$-graph - that is, a $3$-graph which has edge-density at least $d$ on all linearly sized subsets - contains a copy of some $F \in \mathcal{F}$. Let $\Pi_{\therefore,\text{fin}}$ denote the set of all possible uniform Tur\'{a}n densities of finite families. Erd\H{o}s, Hajnal, and R\"{o}dl introduced a family of constructions for lower bounds on uniform Tur\'an densities called palette constructions. We show that $\Pi_{\therefore,\text{fin}}$ contains every $d$ that is obtained as the uniform density of an optimized palette construction. A corollary of this is that $\Pi_{\therefore,\text{fin}}$ contains the set of Lagrangians of $3$-graphs and includes irrational numbers. Our work complements a recent result of Lamaison, which states that every value in $\Pi_{\therefore,\text{fin}}$ can be approximated by uniform densities of palette constructions.
An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstra\"ete introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a variant on the classical Tur\'an problem, asking for the maximum number of edges in a $n$-vertex properly edge-colored graph which does not contain a rainbow-$F$. In the following years many authors have studied the asymptotic behavior of $\mathrm{ex}^*(n,F)$ when $F$ is bipartite. In the particular case that $F$ is a tree $T$, the infamous Erd\"os-S\'os conjecture says that the extremal number of $T$ depends only on the size of $T$ and not its structure. After observing that such a pattern cannot hold for $\mathrm{ex}^*$ in the usual setting, we propose that the relative rainbow extremal number $\mathrm{ex}^*(Q_n,T)$ in the $n$-dimensional hypercube $Q_n$ will satisfy an Erd\"os-S\'os type Conjecture and verify it for some infinite families of trees $T$.
The study of uniform Tur\'an densities was initiated in the 1980s by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density of $F$, $\pi_{\therefore}(F)$, is defined as the infimum $d\in[0,1]$ such that every $3$-graph $H$ in which every linearly sized $S\subseteq V(H)$ induces at least $(d+o(1))\binom{\vert S\vert}{3}$ edges must contain a copy of $F$. Disproving Erd\H{o}s's famous jumping conjecture, Frankl and R\"odl showed that the set of Tur\'an densities is not well-ordered. We prove an analogous result for the uniform Tur\'an density, namely that the set $\Pi^{(3)}_{\therefore,\infty}=\{\pi_{\therefore}(\mathcal{F}) : \mathcal{F}\text{ a family of }3\text{-graphs} \}$ is not well-ordered. This is a consequence of a more general result, which in particular implies that for every Lagrangian $\Lambda$ of a $3$-graph and integer $1 \leq t \leq 6$ we have $\frac{t}{6}\Lambda\in \Pi^{(3)}_{\therefore,\infty}$.
The online ordered Ramsey game is played between two players, Builder and Painter, on an infinite sequence of vertices with ordered graphs $(G_1,G_2)$, which have linear orderings on their vertices. On each turn, Builder first selects an edge before Painter colors it red or blue. Builder's objective is to construct either an ordered red copy of $G_1$ or an ordered blue copy of $G_2$, while Painter's objective is to delay this for as many turns as possible. The online ordered Ramsey number $r_o(G_1,G_2)$ is the number of turns Builder takes to win in the case that both players play optimally. Few lower bounds are known for this quantity. In this paper, we introduce a succinct proof of a new lower bound based on the maximum left- and right-degrees in the ordered graphs. We also upper bound $r_o(G_1,G_2)$ in two cases: when $G_1$ is a cycle and $G_2$ a complete bipartite graph, and when $G_1$ is a tree and $G_2$ a clique.
For integers $r \geq 3$ and $t \geq 2$, an $r$-uniform $t$-daisy $\mathcal{D}^t_r$ is a family of $\binom{2t}{t}$ $r$-element sets of the form $$\{S \cup T \ : T\subset U, \ |T|=t \}$$ for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S \cap U = \emptyset$. It was conjectured by Bollob\'as, Leader and Malvenuto (and independently Bukh) that the Tur\'an densities of $t$-daisies satisfy $\lim\limits_{r \to \infty} \pi(\mathcal{D}_r^t) = 0$ for all $t \geq 2$; this has become a well-known problem, and it is still open for all values of $t$. In this paper, we give lower bounds for the Tur\'an densities of $r$-uniform $t$-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers $m \geq 2t \geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $\mathbb{Z}/m\mathbb{Z}$ such that for any $x \in \mathbb{Z}/m\mathbb{Z}$ and any $2t$-element subset $X$ of $\mathbb{Z}/m\mathbb{Z}$, there are $t$ distinct elements of $X$ whose sum is not in the translate $x+R$? This is a slice-analogue of the extremal Hilbert cube problem considered by Gunderson and R\"odl and its generalization studied by Cilleruelo and Tesoro.
Given a finite set of integers $A$, its sumset is $A+A:= \{a_i+a_j \mid a_i,a_j\in A\}$. We examine $|A+A|$ as a random variable, where $A\subset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$ with a fixed probability $p \in (0,1)$. Recently, Martin and O'Bryant studied the case in which $p=1/2$ and found a closed form for $\mathbb{E}[|A+A|]$. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for $\text{Var}(|A+A|)$ and bounds on the number of missing sums in $A+A$, $m_{n\,;\,p}(k) := \mathbb{P}(2n-1-|A+A|=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $\mathbb{E}[|A+A|]$ and $\text{Var}(|A+A|)$ for all $p\in (0,1)$ and establish good bounds for $\mathbb{E}[|A+A|]$ and $m_{n\,;\,p}(k)$. We continue to investigate $m_{n\,;\,p}(k)$ by studying $m_p(k) = \lim_{n\to\infty}m_{n\,;\,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and O'Bryant proved that, for $p=1/2$, $m_{1/2}(6)>m_{1/2}(7) m_{p}(1)<m_{p}(2)$. Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and O'Bryant to correlated sumsets $A+B$ where $B$ is correlated to $A$ by the probabilities $\mathbb{P}(i\in B \mid i\in A) = p_1$ and $\mathbb{P}(i\in B \mid i\not\in A) = p_2$. We provide some preliminary results using the extension of this framework.
In a 2019 paper, Barnard and Steinerberger show that for $f\in L^1(\mathbf{R})$, the following autocorrelation inequality holds: \begin{equation*} \min_{0 \leq t \leq 1} \int_\mathbf{R} f(x) f(x+t)\ \mathrm{d}x \ \leq\ 0.411 ||f||_{L^1}^2, \end{equation*} where the constant $0.411$ cannot be replaced by $0.37$. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics where some problems, such as those of minimal difference basis, can be encapsulated by a convolution inequality similar to the above integral. Barnard and Steinerberger suggest that future research may focus on the existence of functions extremizing the above inequality (which is itself related to Brascamp-Lieb type inequalities). We show that for $f$ to be extremal under the above, we must have \begin{equation*} \max_{x_1 \in \mathbf{R} }\min_{0 \leq t \leq 1} \left[ f(x_1-t)+f(x_1+t) \right] \ \leq\ \min_{x_2 \in \mathbf{R} } \max_{0 \leq t \leq 1} \left[ f(x_2-t)+f(x_2+t) \right] . \end{equation*} Our central technique for deriving this result is local perturbation of $f$ to increase the value of the autocorrelation, while leaving $||f||_{L^1}$ unchanged. These perturbation methods can be extended to examine a more general notion of autocorrelation. Let $d,n \in \mathbb{Z}^+$, $f \in L^1$, $A$ be a $d \times n$ matrix with real entries and columns $a_i$ for $1 \leq i \leq n$, and $C$ be a constant. For a broad class of matrices $A$, we prove necessary conditions for $f$ to extremize autocorrelation inequalities of the form \begin{equation*} \min_{ \mathbf{t} \in [0,1]^d } \int_{\mathbf{R}} \prod_{i=1}^n\ f(x+ \mathbf{t} \cdot a_i)\ \mathrm{d}x\ \leq\ C ||f||_{L^1}^n. \end{equation*}
We study the problem of crescent configurations, posed by Erd\H{o}s in 1989. A crescent configuration is a set of $n$ points in the plane such that: 1) no three points lie on a common line, 2) no four points lie on a common circle, 3) for each $1 \leq i \leq n - 1$, there exists a distance which occurs exactly $i$ times. Constructions of sizes $n \leq 8$ have been provided by Liu, Pal\'{a}sti, and Pomerance. Erd\H{o}s conjectured that there exists some $N$ for which there do not exist crescent configurations of size $n$ for all $n \geq N$. We extend the problem of crescent configurations to general normed spaces $(\mathbb{R}^2, \| \cdot \|)$ by studying strong crescent configurations in $\| \cdot \|$. In an arbitrary norm $\|\cdot \|$, we construct a strong crescent configuration of size 4. We also construct larger strong crescent configurations in the Euclidean, taxicab, and Chebyshev norms, of sizes $n \leq 6$, $n \leq 8$, and $n \leq 8$ respectively. When defining strong crescent configurations, we introduce the notion of line-like configurations in $\|\cdot \|$. A line-like configuration in $\|\cdot \|$ is a set of points whose distance graph is isomorphic to the distance graph of equally spaced points on a line. In a broad class of norms, we construct line-like configurations of arbitrary size. Our main result is a crescent-type result about line-like configurations in the Chebyshev norm. A line-like crescent configuration is a line-like configuration for which no three points lie on a common line and no four points lie on a common $\|\cdot \|$ circle. We prove that for $n \geq 7$, every line-like crescent configuration of size $n$ in the Chebyshev norm must have a rigid structure. Specifically, it must be a perpendicular perturbation of equally spaced points on a horizontal or vertical line.
Let $\ell_1,\ell_2,\dots$ be a countable collection of lines in ${\mathbb R}^d$. For any $t \in [0,1]$ we construct a compact set $\Gamma\subset{\mathbb R}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each $\ell_i$, such that the image of each projection has dimension $t$. This immediately implies the existence of homeomorphisms between certain Cantor-type sets whose graphs have large dimensions. As an application, we construct a collection $E$ of disjoint, non-parallel $k$-planes in $\mathbb{R}^d$, for $d \geq k+2$, whose union is a small subset of $\mathbb{R}^d$, either in Hausdorff dimension or Lebesgue measure, while $E$ itself has large dimension. As a second application, for any countable collection of vertical lines $w_i$ in the plane we construct a collection of nonvertical lines $H$, so that $F$, the union of lines in $H$, has positive Lebesgue measure, but each point of each line $w_i$ intersects at most one $h\in H$ and, for each $w_i$, the Hausdorff dimension of $F\cap w_i$ is zero.