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# High-dimensional combinatorics

## Final Report Summary - HI-DIM COMBINATORICS (High-dimensional combinatorics)

Let us recall the basic concepts that a student sees in a first undergraduate class of discrete mathematics. The list includes things like graphs, trees, permutations, tournaments etc. When viewed from a geometric perspective it is very clear that all these objects are one-dimensional. This research project starts from a very general question - Do these objects have higher-dimensional counterparts? If so, are the higher dimensional phenomena essentially the same or is the behavior fundamentally different? Perhaps the easiest example with which to demonstrate these ideas is to look at graphs. For a topologist a graph is just a one-dimensional simplicial complex. So, it is obvious what to ask: Take any set of ideas, insights or phenomena from graph theory and investigate how they materialize in the realm of higher dimensional simplicial complexes. For example: One of the main subareas of modern graph theory is the field of random graphs. It is a field with a rich mathematical theory, and with a wealth of applications in many different areas. As Erdos and Renyi discovered nearly 60 years ago, there is a well-defined and explicitly calculated critical edge density at which the graph becomes connected. The analogous higher-dimensional question is substantially more difficult and was answered by Linial and meshulam over a decade a go. However, the analog of Erdos and Renyi’s most remarkable discovery has remained unknown. They showed that there is a critical density at which a random graphs acquires a giant component. Also, at the same critical density the graph almost surely ceases to be a forest. This illustrates nicely some of the conceptual difficulties that we had encountered: (i) There are several natural ways to define what a forest is in higher dimensions. Which one should we consider. Answer: all of them. In higher dimensions these lead to separate fascinating phenomena. (ii) There is no high-dimensional notion of a connected component. What to do? Answer: Find a substitute (what we call “giant shadow”, which in one dimension essentially coincides with the notion of a giant component and does extend smoothly to higher dimensions. This research project has discovered fundamental properties of higher dimensional permutations and trees. High-dimensional combinatorics has now become an established subfield of discrete mathematics in which many leading investigators show great interest.