Nested cycles without crossings — k-cycle version

Establish the existence of a function d_0(k) such that, for every integer k ≥ 3, every graph G with average degree at least d_0(k) contains a sequence C_1, …, C_k of nested cycles with no crossings (i.e., the cyclic order of C_{i+1} respects that of C_i).

Background

Gil Fernández, Kim, Kim, and Liu proved the existence of two nested cycles without crossings for sufficiently large average degree. Extending this to k nested cycles would generalise Erdős’s nested-cycle phenomena to longer sequences while preserving the no-crossing structure.

References

Question Does there exist d_0(k) such that, for every k ≥ 3, every graph G with d(G) ≥ d_0(k) has a sequence C_1, …, C_k of nested cycles with no crossings?

— Sublinear expanders and their applications  (2401.10865 - Letzter, 2024) in Nested cycles (Section 8.1)

A layering lemma with a degree requirement of order $r\log r$ instead of $r2\log r$ would allow the weight $\Lambda(n){1+\varepsilon}$ for any fixed $\varepsilon>0$. It would then give $f_k(n)=O_{k,\varepsilon}\bigl(n(\log\log n){1+\varepsilon}\bigr)$. We do not know whether such a lemma holds.

— An $n(\log n)^{o(1)}$ bound for nested cycles without geometric crossings  (2609.02234 - Ai et al., 2 Sep 2026) in Section 6, “Concluding remarks,” subsection “Limits of the method”