Mubayi–Suk Conjecture on f^{(4)}_{5,6}(N)

Establish that the generalized Erdős–Rogers function f^{(4)}_{5,6}(N)—defined as the largest integer m such that every K^{(4)}_6-free 4-uniform hypergraph on N vertices contains an induced K^{(4)}_5-free subgraph on m vertices—satisfies f^{(4)}_{5,6}(N) = (log log N)^{Θ(1)}.

Background

The paper studies generalized Erdős–Rogers functions f^{(k)}_{F,G}(N), focusing on the case F=K^{(4)}_5 and G=K^{(4)}_6. Mubayi and Suk conjectured that f^{(4)}_{5,6}(N) grows like a fixed power of log log N. This k=4, s=6 case is the pivotal remaining instance in a broader program to determine f^{(k)}_{k+1,k+2}(N) up to polylogarithmic factors.

The authors prove the analogous order of magnitude for the near-clique 5^{-} (K^{(4)}_5 with one edge deleted), showing f^{(4)}_{5^{-},6}(N)=(log log N)^{Θ(1)}, and extend this via a stepping-up framework to related functions. However, the exact K^{(4)}_5 case remains conjectural, and resolving it would settle the k=4 base case needed for general stepping-up arguments toward f^{(k)}_{k+1,k+2}(N).