Randomness and structure in combinatorics

Introduction

Randomness plays an inseparable role in combinatorics. Indeed, non-constructive probabilistic arguments are a powerful way to prove the existence of various kinds of combinatorial objects, and the study of random discrete structures has illuminated nearly all fields of combinatorics. I propose a program to achieve a deeper understanding of this role of randomness in combinatorics, emphasising the relationship between “structured” (hence explicit) objects, and random or “random-like” objects.

Investigation of Ramsey Graphs

A) The Role of Probabilistic Arguments

There are many situations in combinatorics where probabilistic arguments demonstrate that “almost all” objects satisfy a certain property, but it is difficult to explicitly specify an object with the property. The most notorious examples are in Ramsey theory, which studies how “disordered” it is possible for an object to be.

Goals

I plan to investigate the structure of Ramsey graphs, with the goals of:

• Unifying the area
• Making decisive progress on important conjectures

Probabilistic Aspects of Design Theory

B) Slower Adoption of Probabilistic Methods

Conversely, certain areas of combinatorics have been slower to benefit from the probabilistic method; particularly areas in which algebraic constructions play a major role.

Design Theory

Design theory is the study of combinatorial “arrangements” with very strong regularity properties, most naturally obtained by exploiting symmetry/regularity properties of algebraic structures.

Research Plans

I plan to investigate probabilistic aspects of design theory, and in particular to:

• Build a theory of random designs

Structure vs Pseudorandomness

C) The Dichotomy

Actually, structure and randomness often come together, due to the “structure vs pseudorandomness dichotomy” elucidated by Tao. Indeed, there are many important problems in combinatorics for which it is known how to solve both random instances and “structured” instances.

Decomposition of Instances

In such cases, we hope to decompose general instances into structured and pseudorandom parts, handled by different means.

Concrete Problems

I describe several concrete problems in this vein, whose study will advance our general understanding of this phenomenon.