The Turán number of the triangular pyramid of 4-layers

The Turán number $ex(n,H)$ of a graph $H$ is the maximum number of edges in any $H$-free graph on $n$ vertices. The triangular pyramid of $k$-layers, denoted by $TP_k$, is a generalization of a triangle. The Turán problems of a triangular pyramid with small layers have been studied widely by Liu (E-JC, 2013), Xiao, Katona, Xiao and Zamora (DAM, 2022), Ghosh, Győri, Paulos, Xiao and Zamora (DAM, 2022). Moreover, Ghosh et al. conjectured that $ex(n, TP_4)=\frac{1}{4}n^2+Θ(n^{\frac{4}{3}})$. In this note, we confirm this conjecture.

💡 Research Summary

The paper addresses the Turán number ex(n, TP₄) for the 4‑layer triangular pyramid graph TP₄, a non‑bipartite graph that generalizes a triangle. Earlier work determined exact Turán numbers for TP₁ (a triangle) and TP₂, and gave asymptotic bounds for TP₃. For TP₄, Ghosh et al. conjectured that ex(n, TP₄)=¼ n² + Θ(n^{4/3}) but only a lower bound matching the Θ‑term was known. This note confirms the conjecture by establishing a matching upper bound.

The authors first define TPₖ: for each i=1,…,k+1 they draw a path P_i with i vertices, then connect consecutive layers by edges y_{i,t}y_{i+1,t} and y_{i,t}y_{i+1,t+1}. Thus TP₁ is K₃, TP₂ has been fully solved, and TP₃’s Turán number is known up to a linear term.

The main result (Theorem 1.6) states that for sufficiently large n, \