For many well-known families of triple systems $\mathcal{M}$, there are perhaps many near-extremal $\mathcal{M}$-free configurations that are far from each other in edit-distance. Such a property is called non-stable and is a fundamental barrier to determining the Turán number of $\mathcal{M}$. Liu and Mubayi gave the first finite example that is non-stable. In this paper, we construct another finite family of triple systems $\mathcal{M}$ such that there are two near-extremal $\mathcal{M}$-free configurations that are far from each other in edit-distance. We also prove its Andrásfai-Erdős-Sós type stability theorem: Every $\mathcal{M}$-free triple system whose minimum degree is close to the average degree of the extremal configurations is a subgraph of one of these two near-extremal configurations. As a corollary, our main result shows that the boundary of the feasible region of $\mathcal{M}$ has exactly two global maxima.
@article{10_37236_11701,
author = {Yixiao Zhang and Jianfeng Hou and Heng Li},
title = {A 2-stable family of triple systems},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {2},
doi = {10.37236/11701},
zbl = {1536.05256},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11701/}
}
TY - JOUR
AU - Yixiao Zhang
AU - Jianfeng Hou
AU - Heng Li
TI - A 2-stable family of triple systems
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11701/
DO - 10.37236/11701
ID - 10_37236_11701
ER -
%0 Journal Article
%A Yixiao Zhang
%A Jianfeng Hou
%A Heng Li
%T A 2-stable family of triple systems
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11701/
%R 10.37236/11701
%F 10_37236_11701
Yixiao Zhang; Jianfeng Hou; Heng Li. A 2-stable family of triple systems. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/11701
