Year: 2025
Type: article
Abstract: For every positive integer t we construct a finite family of triple systems $${\cal M}_{t}$$ M t , determine its Turán number, and show that there are t extremal $${\cal M}_{t}$$ M t -free configurations that are far from each other in edit-distance. We also prove a strong stability theorem: every $${\cal M}_{t}$$ M t -free triple system whose size is close to the maximum size is a subgraph of one of these t extremal configurations after removing a small proportion of vertices. This is the first stability theorem for a hypergraph problem with an arbitrary (finite) number of extremal configurations. Moreover, the extremal hypergraphs have very different shadow sizes (unlike the case of the famous Turán tetrahedron conjecture). Hence a corollary of our main result is that the boundary of the feasible region of $${\cal M}_{t}$$ M t has exactly t global maxima.
Source: Israel Journal of Mathematics
Authors: Xizhi Liu, Dhruv Mubayi, Christian Reiher
Institutions: University of Illinois Chicago, Universität Hamburg
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: National Science Foundation
OpenAlex ID: W4411958960