arXiv subjects
Publications and source records attributed to Kada Williams.
In response to a conjecture of Petr and Turek, we prove that every largest intersecting temperate collection of sets $\mathcal{A}\subseteq [2k]^{(k)}\cup [2k]^{(k+1)}\cup [2k]^{(k+2)}$ includes all $k$-sets containing some given $a\in[2k]$, all $(k+1)$-sets, and all $(k+2)$-sets not containing $a$.
In 1983, Daykin and Frankl conjectured that if $P$ is a convex subset of $Q_n$, then it contains at least $|P|\binom{n}{\lfloor n/2\rfloor}2^{-n}$ pairwise incomparable elements. We verify and communicate an LLM-generated proof of this conjecture.
Define a queen on $\mathbb{Z}_n^d$ with admissible moves parallel to $\mathbf{x}\in\{-1,0,1\}^d$, of arbitrary length. How many queens can be placed on $\mathbb{Z}_n^d$ without any two in conflict? In two dimensions, this problem was initiated by Pólya in 1918 and resolved by Monsky in 1989. We give the first known impossibility result in $d>2$ dimensions, showing that a trivial upper bound $n^{d-1}$ cannot be achieved if $n$ is a multiple of $5$ and not of $25$. Moreover, we conjecture that $n^{d-1}-O(n^{d-2})$ queens can be placed independently, which we prove if $n$ has no prime divisor less than $2^{\lfloor d/2\rfloor +1}$.
The renowned Gossiping Problem (1971) asks the following. There are $n$ people who each know an item of gossip. In a telephone call, two people share all the gossip they know. How many calls are needed for all of them to be informed of all the gossip? If $n\ge 4$, the answer is $2n-4$. We initiate and solve the related Greedy Gossiping Problem: given a fixed number $m<2n-4$ of calls, at most how much gossip can be known altogether? If every call increases the total knowledge of gossip as much as possible, the sum reaches $n^2$ only when $m=2n-3$. Our main result is that surprisingly, for each $m<2n-4$, this calling strategy is optimal.
Let $X$ and $Y$ be upward closed set systems in the lattice of $\{0,1\}^n$. The celebrated Harris-Kleitman inequality implies that if $|X|=α2^n$, $|Y|=β2^n$, the density of the set of points in exactly one of $X$ and $Y$ is maximal when $X$ and $Y$ are independent, meaning $|X\cap Y|=αβ2^n$. Is the same true of three upward closed systems, $X$, $Y$, and $Z$? Suppose $|X|=|Y|=|Z|$. Kahn asked whether the set of points in exactly one of $X$, $Y$, $Z$ has density at most $\frac49$. We answer this question in the negative.
The width of a poset is the size of its largest antichain. Sperner's theorem states that $(2^{[n]},\subset)$ is a poset whose width equals the size of its largest layer. We show that Hamming ball posets also have this property. This extends earlier work that proves this in the case of small radii. Our proof is inspired by (and corrects) a result of Harper.
Fernández-Bretón, Sarmiento and Vera showed that whenever a direct sum of sufficiently many copies of ${\mathbb Z}_4$, the cyclic group of order 4, is countably coloured there are arbitrarily large finite sets $X$ whose sumsets $X+X$ are monochromatic. They asked if the elements of order 4 are necessary, in the following strong sense: if $G$ is an abelian group having no elements of order 4, is it always the case there there is a countable colouring of $G$ for which there is not even a monochromatic sumset $X+X$ with $X$ of size 2? Our aim in this short note is to show that this is indeed the case.
A subgraph of the $n$-dimensional hypercube is called 'layered' if it is a subgraph of a layer of some hypercube. In this paper we show that there exist subgraphs of the cube of arbitrarily large girth that are not layered. This answers a question of Axenovich, Martin and Winter. Perhaps surprisingly, these subgraphs may even be taken to be induced.
Various authors have calculated how many pairwise incomparable points can be selected from a partially ordered set. We tackle this question for the family of subsets of a finite set obtained by removing or adding a bounded number of elements from a given subset. Our versatile approach is proven valid under the condition of the set size exceeding the cube of the radius.
Consider two regions in the plane, bounded by an $n$-gon and an $m$-gon, respectively. At most how many connected components can there be in their intersection? This question was asked by Croft. We answer this asymptotically, proving the bounds $$\left\lfloor \frac{m}{2}\right\rfloor \cdot \left\lfloor \frac{n}{2}\right\rfloor\le f(n,m)\le \left\lfloor \frac{m}{2}\right\rfloor \cdot \frac{n}{2} + \frac{m}{2} $$ where $f(n,m)$ denotes the maximal number of components and $m\le n$. Furthermore, we give an exact answer to the related question of finding the maximal number of components if the $m$-gon is required to be convex: $\left \lfloor \frac{m+n-2}{2}\right\rfloor$ if $n\ge m+2$ and $n-2$ otherwise.