Immediate RIch Structures

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.