A single extra connection in a vast network can flip a property from impossible to almost certain—mathematicians call this moment the threshold, and they are learning to predict it with precision. This project tackles a blind spot in network theory. For decades, researchers could only estimate where thresholds occur, often missing the exact tipping point. Recent breakthroughs solved the Kahn-Kalai conjecture, but left two gaps: the precise threshold for properties like bounded-degree spanning trees remains unknown, and what happens *immediately after* the threshold—the sudden richness of new structures—is poorly understood. The work aims to turn rough estimates into exact answers for properties triggered by local events, and to build a universal framework for studying network behaviour right at the moment of emergence. This is fundamental mathematics, not applied engineering. It will not directly change a power grid or a social media algorithm tomorrow. But random graphs underpin everything from internet routing to epidemic modelling. Knowing exactly when and how complex structures appear—rather than roughly where—could eventually sharpen predictions in any system that relies on networks. Past work on graph thresholds has already reshaped how we understand phase transitions; this project pushes that understanding to the next level of detail.
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Random graph exploration, initiated by Erdos and Rényi, aids in understanding typical graph characteristics and their representation in complex network models. They spotlighted "thresholds" in graph properties, describing a foundational phenomenon where a property shifts from being improbable to likely due to a minor alteration in an underlying density parameter. For decades, extensive research has focused on determining the precise location of these thresholds for diverse graph properties. Until recently, threshold estimations were generally carried out in an ad-hoc manner, often involving intricate graph-theoretic arguments. A pivotal breakthrough was made by Frankston, Kahn, Narayanan, and Park, followed by Park and Pham. They provided a universal tool to locate thresholds up to logarithmic factors, resolving the celebrated Kahn-Kalai conjecture. Their methodology, though revolutionary, has two main limitations. First, it falls short in pinpointing the exact location of the threshold when it is sharp, leaving essential questions unanswered, such as the precise threshold for the emergence of bounded-degree spanning trees. Second, it offers only limited insights into the richness of the property immediately after its emergence. This project's objectives are to: (i) Identify exact thresholds for properties whose emergence is tied to "local" events, (ii) adapt recent breakthroughs, crafting a comprehensive framework to study the extremal richness of these properties post-emergence. While specific important cases, like bounded-degree spanning trees and triangle factors, will be closely examined, the ultimate aspiration is to develop a universal methodology. Based at the University of Oxford and guided by Prof. Keevash, this project targets foundational challenges in random graph theory. Combining the fellow's innovative strategies with expert supervision, the endeavour is poised to pioneer new avenues, setting the stage for future investigations in this field.
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