Asymptotics of Turán numbers for complete-graph suspensions

Determine the asymptotic behavior of ex_r(n,S_rK_k) for r-uniform suspensions of complete graphs K_k, beyond the cases currently known.

Background

For a graph F, the r-uniform suspension S_rF is formed by adding r−2 common new vertices to every edge of F. Theorem 1.2 reduces the asymptotic study of suspensions of graphs with chromatic number k to the corresponding suspension of K_k. However, the paper notes that the relevant asymptotics are largely unknown, with the exception of the case r=4 and k=3.

References

For larger uniformity, not many asymptotic results are implied, since we do not even know the asymptotics of exr(n, Sr Kk), with the exception of r = 4, k = 3.

— On Turán problems for suspension hypergraphs  (2502.10905 - Cheng et al., 15 Feb 2025) in Section 1, Introduction, p. 3