An Andrásfai–Erdős–Sós type theorem for F3,3 in the ℓ2-norm

Let F 3 , 3 be the 3-uniform hypergraph on six vertices with edge set { 123 , 145 , 146 , 156 , 245 , 246 , 256 , 345 , 346 , 356 } . In this note, we establish an Andrásfai–Erdős–Sós type stability theorem for F 3 , 3 in the ℓ 2 -norm: There exists a positive constant ξ such that for all sufficiently large n , every F 3 , 3 -free 3-uniform hypergraph on n vertices with minimum ℓ 2 -norm degree at least ( 5 / 4 − ξ ) n 3 must be bipartite. This strengthens a result of Balogh, Clemen, and Lidický [3] .