Extremal Graphs and Hypergraphs

Graph theory can trace its roots back to the work of Euler on the problem of the bridges of Königsberg in 1736. It has assumed a growing importance within both pure and applied mathematics since then, especially in the last century due to the development of computer science.

Graphs are the mathematical objects used to represent pairwise relations within a set, and as such are essential to the study of any network-based phenomenon, for example, communication networks, transport logistics, the spread of diseases or information in human societies, the performance of networkbased algorithms or the architecture of the internet. Within mathematics, graphs and their generalisation to hypergraphs play a key role in additive number theory, probability theory and group theory, in addition to being studied for their own sake.

This wealth of applications has led to an increased dynamism and vitality in the eld, which has grown tremendously in recent decades, and resulted in a number of landmark achievements in recent years, such as the proof of the existence of designs conjecture, the development of the hypergraph container method, and 2023’s succession of stunning breakthroughs on diagonal and asymmetric Ramsey numbers.

The goal of this workshop is to gather thirty researchers in extremal graph and hypergraph theory to discuss the latest development in the eld, the direction research efforts should take next (which old conjectures that were out of reached for decades are now ripe for renewed assaults? what are the key obstacles to progress?) and to collaborate intensely for a week while at the institute.

The workshop will involve a diverse mix of senior, mid-career and junior participants so as to maximise the creativity and collaborative potential of the group as a whole. The intended workshop design will also leave at least half of the time at the institute free for work in smaller constellations, with some “big picture” talks, surveys of new problems and techniques interspersed throughout the week providing a framework around which collaborations can develop organically.