The exact minimum number of triangles in graphs of given order and size

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s in 1955; it is now known as the Erd\H{o}s-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from~$1$, which in this range confirms a conjecture of Lov\'asz and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.Comment: Published in Forum of Mathematics, Pi, Volume 8, e8 (2020

Proof of Koml\'os's conjecture on Hamiltonian subsets

Koml\'os conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ

Rethinking Causation in Cancer through Modularity

Despite the productivity of basic cancer research, cancer continues to be a health burden to society because this research has not yielded corresponding clinical applications. Many proposed solutions to this dilemma have revolved around implementing organizational and policy changes related to cancer research. Here I argue for a different solution: a new conceptualization of causation in cancer. Neither the standard molecular biomarker approaches nor evolutionary medicine approaches to cancer fully capture its complex causal dynamics, even when considered jointly. These approaches map on to Ernst Mayr’s proximate-ultimate distinction, which is an inadequate conceptualization of causation in biological systems and makes it difficult to connect developmental and evolutionary viewpoints. I conceptualize causation in cancer through the notion of modularity as a bridge between molecular biomarker approaches and evolution medicine approaches. A modularity-based approach requires the consideration of relationships between multiple levels of organization and the incorporation of different time scales, thereby overcoming the proximate-ultimate divide. The proposed perspective on causation in cancer is better suited to integrating the complexity of current empirical results and can facilitate novel developments in the investigation and clinical treatment of cancer

Rethinking Causation in Cancer with Evolutionary Developmental Biology

Despite the productivity of basic cancer research, cancer continues to be a health burden to society because this research has not yielded corresponding clinical applications. Many proposed solutions to this dilemma have revolved around implementing organizational and policy changes related to cancer research. Here I argue for a different solution: a new conceptualization of causation in cancer. Neither the standard molecular biomarker approaches nor evolutionary biology approaches to cancer fully capture its complex causal dynamics, even when considered jointly. These approaches map on to Ernst Mayr’s proximate-ultimate distinction, which is an inadequate conceptualization of causation in biological systems and makes it difficult to connect developmental and evolutionary viewpoints. I propose looking to evolutionary developmental biology (evo devo) to overcome the distinction and integrate the proximate and ultimate causal frameworks. I use the concepts of modularity and evolvability to show how an evo devo perspective can be manifested in cancer translational research. This perspective on causation in cancer is better suited for integrating the complexity of current empirical results and can facilitate novel developments in the investigation and clinical treatment of cancer

Slow nucleation rates in Chain Inflation with QCD Axions or Monodromy

The previous proposal (by two of us) of chain inflation with the QCD axion is shown to fail. The proposal involved a series of fast tunneling events, yet here it is shown that tunneling is too slow. We calculate the bubble nucleation rates for phase transitions in the thick wall limit, approximating the barrier by a triangle. A similar problem arises in realization of chain inflation in the string landscape that uses series of minima along the monodromy staircase around the conifold point. The basic problem is that the minima of the potential are too far apart to allow rapid enough tunneling in these two models. We entertain the possibility of overcoming this problem by modifying the gravity sector to a Brans-Dicke theory. However, one would need extremely small values for the Brans-Dicke parameter. Many successful alternatives exist, including other "axions" (with mass scales not set by QCD) or potentials with comparable heights and widths that do not suffer from the problem of slow tunneling and provide successful candidates for chain inflation.Comment: 6 pages, 1 figur

Local conditions for exponentially many subdivisions

Given a graph F, let st(F) be the number of subdivisions of F, each with a different vertex set, which one can guarantee in a graph G in which every edge lies in at least t copies of F. In 1990, Tuza asked for which graphs F and large t, one has that st(F) is exponential in a power of t. We show that, somewhat surprisingly, the only such F are complete graphs, and for every F which is not complete, st(F) is polynomial in t. Further, for a natural strengthening of the local condition above, we also characterize those F for which st(F) is exponential in a power of t

Foreword to the EEGSA-SEGSA Conference Proceeding: Exploring Horizons

The 14th Annual Graduate Student Showcase was held by the Department of Secondary and Elementary Education on April 30, 2022. The COVID-19 pandemic has continued to impact the way in which we gather and share our research. This year, we continued to meet in a hybrid synchronous/asynchronous space with our keynote speakers, presenters, and attendees. This format allowed for graduate students from around the world to share and participate.  This year, our showcase theme, Exploring Horizons, speaks to looking forward, keeping our heads up, and navigating what lies beyond. Despite numerous challenges to this academic year, as a graduate student community, this research showcase demonstrated that graduate student research in education is attuned to the possibilities that might lie on or on the other side of the horizons.  Exploring Horizons also represents an invitation to think and rethink differently and creatively. Currently, the field of education is in the midst of dealing with some serious challenges. Yet, when we simply glance at this showcase special issue, it is easy to see that our graduate students in education are rising to the occasion and exploring the horizons of educational research, ready to take on these challenges and to open new possibilities for imagining the future(s) of curriculum, pedagogy, teacher education, and educational philosophy and thought. This year’s theme of Exploring Horizons asks participants and attendees to visualize and imagine the unknown and all that becomes possible by thinking, creating, and imagining new ways of knowing and understanding. To all of our authors in this special issue, we want to thank you for putting your important work out into the world and for sharing it with others. Additionally, this special issue Exploring Horizons celebrates the diversity of our graduate student research in education.  Beyond presentations from graduate students, we featured Dr. Cathryn van Kessel, a keynote speaker in the morning session and an associate professor for the Department of Secondary Education, as she shared her experiences with “Exploring Horizons” in knowledge dissemination and research mobilization. Dr. Jacqueline Filipek, an alumna of the Department of Elementary Education and assistant professor at The King’s University, shared valuable insights about the process for applying and interviewing for academic jobs in the afternoon session. She offered generous advice on how graduate students might prepare their application packages and think about potential interview processes in academia. On behalf of the EEGSA/SEGSA Graduate Student Research Showcase Committee, Katherine Koskie, Kimberly Edmondson, and Yina Liu, we wish to thank you for reading this special issue as you explore new horizons together with our authors. We hope you find your reading thought-provoking as you consider what is on the horizon of educational research

The role of microRNAs in acute myeloid leukemia

MicroRNAs (miRs) are short (18-22 nucleotides) non-coding RNAs that are important in regulating gene expression. MiR expression is deregulated in many types of cancers, including leukemias. In acute myeloid leukemia (AML), the expression of specific miRs has been linked with both prognostically and cytogenetically defined subgroups. Recent studies have shown that deregulation of miR expression is not simply a consequence of AML but a potential contributer to leukemogenesis. This commentary will focus on select findings that describe the different mechanistic roles for miRs in the development of leukemia