Problems and results on linear hypergraphs

In this thesis, we tackle several problems involving the study of 3-uniform, linear hypergraphs satisfying some additional structural constraint. We begin with a problem of Hrushovski concerning Latin squares satisfying a partial associativity condition. From an $n\times n$ Latin square $A$ one can define a binary operation $\circ:[n]\times[n]\to [n]$, and $\circ$ is associative if and only if $A$ is a group multiplication table. Hrushovski asked whether, if $\circ$ is only associative a positive proportion of the time, $A$ must still in some sense be close to a group multiplication table. This problem manifests a well-studied combinatorial theme, in which a local structural constraint is relaxed (first to a `99$\%$' version and then to a `1$\%$' version) and the global consequences of the relaxed constraints are analysed. We show that the partial associativity condition is sufficient to deduce powerful global information, allowing us to find within $A$ a large subset with group-like structure. Since Latin squares can be regarded as 3-uniform, linear hypergraphs, and the partial associativity condition can be formulated in terms of the count of a particular subhypergraph, we are able to apply purely combinatorial methods to a problem that touches algebra, model theory and geometric group theory. We then take this problem further. A condition due to Thomsen provides a combinatorial constraint which, if satisfied by the Latin square $A$, proves that $A$ is in fact the multiplication table of an abelian group. It is then natural to ask whether a relaxed version of this result is also attainable, and by extending our methods we are able to prove a result of this flavour. Since the combinatorial obstructions to commutativity of $\circ$ are far more complex than those for associativity, topological complications arise that are not present in the earlier work. We also study a problem of Loh concerning sequences of triples of integers from $[n]$ satisfying a certain `increasing' property. Loh studied the maximum length of such a sequence, improving a trivial upper bound of $n^2$ to $n^2/\exp(\log^*n)$ using the triangle removal lemma and conjecturing that a natural construction of length $n^{3/2}$ is best possible. We provide the first power-type improvement to the upper bound, showing that there exists $\epsilon>0$ such that the length is bounded by $n^{2-\epsilon}$. By viewing the triples as edges in a 3-uniform hypergraph, the increasing property shows that the hypergraph is linear and provides further restrictions in terms of forbidden subhypergraphs. By considering this formulation, we provide links to various important open problems including the Brown--Erd\H os--S\'os conjecture. Finally, we present a collection of shorter results. In work connecting to the earlier chapters, we resolve the Brown--Erd\H os--S\'os conjecture in the context of hypergraphs with a group structure, and show moreover that subsets of group multiplication tables exhibit local density far beyond what can be hoped for in general. In work less closely connected to the main theme of the thesis, we also answer a question of Leader, Mili\'cevi\'c and Tan concerning partitions of boxes, consider a problem on projective cubes in $\mathbb{Z}_{2^n}$, and resolve a conjecture concerning a diffusion process on graphs

Hollins Columns (1940 Nov 28)

Table of Contents: 1940 Christmas Pageant Presents Madonna: Ye Merry Masquers Select One of Three Plays By Hollins Students—Virginia Dean Talks on Biology: Ivey Lewis Lectures—Broken Heart May Be Soothed by Sophomore’s New Jilt Club—Executive Council Considers Policy: Plans Underway Seek to Improve Academic Life on Campus—Divisions Meet, Discuss Courses: Changes in Curriculum Suggested to Committee—Program of Christmas Organ Music Planned—Couper Elected Treasurer of Student Government—Delegates Attend I.R.C. Convention—Hollins Cotillion Holds Christmas Dance Tomorrow—Edward Ellsberg, Naval Authority, Speaks Here on World Affairs: Noted Authority and Scientist Talks to Campus Community on Defense of America—Banquet, Turkey Dinner Highlight Brief Holiday—Hollins Columns Staff—Letters to the Editor—Under the Dome—The Clothes Bag—Fraser Addresses Honor Students—Choral Club Group Hits One Flat—Thomas Leads Evens to Overwhelming Victory—Organizations—Schaeffer Speaks on Egyptian Art—Five Easy Lessons on How to Study—Sport Slantshttps://digitalcommons.hollins.edu/newspapers/1155/thumbnail.jp

The Hetero Home: The Scenic Design of Straight White Men For the Royall Tyler Theatre

The play Straight White Men by Young Jean Lee emphasizes challenges to established suburban norms of straight white masculinity, what it means to be a family, and how it feels to be home for the holidays. In my scenic design for the Royall Tyler Theatre’s production in 2022, I used personal experience as a middle-class, queer white woman to shape the familiar suburban living room into a simulated, queered display to explore empathizing with the straight white male characters of the show, and those who the audience may know in real life. My fellow collaborators and I had the opportunity to explore identity and suburbia, build a space which is often exclusionary and imbue it with care, queerness, and empathy. Through the construction of a heterosexual home, I discovered things about scenic design, myself as an artist, and the importance of representation in cultivating healthy creative spaces for non-straight, non-white, and/or non-men

Interview with Russ McGibney

Russ McGibney , owner of Sips Coffee House and Deli, talks about growing up in Mount Vernonhttps://digital.kenyon.edu/ps_interviews/1023/thumbnail.jp

The Frontier, March 1927

This is volume 7, number 2.https://scholarworks.umt.edu/frontier/1019/thumbnail.jp

I Am Error

I Am Error is a platform study of the Nintendo Family Computer (or Famicom), a videogame console first released in Japan in July 1983 and later exported to the rest of the world as the Nintendo Entertainment System (or NES). The book investigates the underlying computational architecture of the console and its effects on the creative works (e.g. videogames) produced for the platform. I Am Error advances the concept of platform as a shifting configuration of hardware and software that extends even beyond its ‘native’ material construction. The book provides a deep technical understanding of how the platform was programmed and engineered, from code to silicon, including the design decisions that shaped both the expressive capabilities of the machine and the perception of videogames in general. The book also considers the platform beyond the console proper, including cartridges, controllers, peripherals, packaging, marketing, licensing, and play environments. Likewise, it analyzes the NES’s extension and afterlife in emulation and hacking, birthing new genres of creative expression such as ROM hacks and tool-assisted speed runs. I Am Error considers videogames and their platforms to be important objects of cultural expression, alongside cinema, dance, painting, theater and other media. It joins the discussion taking place in similar burgeoning disciplines—code studies, game studies, computational theory—that engage digital media with critical rigor and descriptive depth. But platform studies is not simply a technical discussion—it also keeps a keen eye on the cultural, social, and economic forces that influence videogames. No platform exists in a vacuum: circuits, code, and console alike are shaped by the currents of history, politics, economics, and culture—just as those currents are shaped in kind

Stories That I\u27ve Heard Before

An episodic narrative about a family from New Jersey, told from the perspective of a mother to her young son and focusing on the decline of her father

Searching For Black Creek: An Experimental Expedition Into Narrative Ecotheory

This paper is an exploration of the land around Black Creek, a polluted urban creek northwest of downtown Toronto. It flows above ground for some 35 kilometers from its source in the southern edge of the Oak Ridges Moraine through industrial areas and low-income neighbourhoods before its eventual confluence with the Humber River. The paper takes the form of a narrative expedition upstream from the creek's mouth. The fictionalized author recounts layers of historical, geographical, and theoretical knowledge in an effort to develop a familiarity with the creek, to understand how it came to be as it is, and to aid in the voyage to its source. At the same time, the paper ambivalently engages with the impossibility of objective knowledge and the consequences of actions driven by anthropocentric, post-Enlightenment ideology. In an effort to overcome this, the narrator attempts to deploy a hybrid of speculative realism and vital materialism to better build an understanding of the creek, this cyborg entity. The paper moves toward a frantic search for methods to engage with or even to understand the catastrophic anthropogenic forces which seem to have developed a life of their own - seeking their own ends without the faintest ecological concern

All We Are is All We Were: the Bridge and Blight of Sandwich Towne

In Windsor, Ontario there is a historic community known as Sandwich Towne that is currently being destroyed by a single man: Matty Moroun. He owns the Ambassador Bridge, the busiest border crossing in North America, as well as a stretch of more than 110 abandoned houses running parallel to the bridge. Moroun plans to knock houses down and building a second bridge to Detroit, despite the fact that the Canadian government is already building their own span. All We Are is All We Were: the Bridge and Blight of Sandwich Towne is a creative nonfiction exploration of this community stricken with urban blight. Through using a combination of prose and comics storytelling, the work not only serves to tell the tale of walking through and talking to residents in Sandwich, but also embodies the conflict between objective and subjective telling of “truth” in a work of creative nonfiction