Improve the linear size threshold for vertex-disjoint triangles

Improve the hypothesis n≥5k+2 in the density criterion for k vertex-disjoint triangles in an n-balanced tripartite graph to n>Ck for every constant C>1, provided n is sufficiently large.

Background

The main theorem proves that a (k,n)-cyclic triple of bipartite edge densities guarantees k vertex-disjoint triangles when n≥5k+2. The conjecture asks whether the required relationship between the part size n and the number k of disjoint triangles can be reduced to an arbitrary linear threshold with constant strictly greater than one.

Such an improvement would extend the theorem to partial triangle-factors with k proportional to n, covering every fixed density of triangles below a full triangle-factor.

References

The bound $n\geq 5k+2$ in Theorem \ref{main-theorem} can be improved to $n>C k$ for any constant $C>1$, for large enough $n$.

— Density conditions for $k$ vertex-disjoint triangles in tripartite graphs  (2503.05218 - Guo et al., 7 Mar 2025) in Conjecture 1, immediately following Remark after Theorem 1

We believe that the condition on the ratio between $n$ and $k$ in Theorem \ref{main-theorem} can be improved. The bound $n\geq 5k+2$ in Theorem \ref{main-theorem} can be improved to $n>C k$ for any constant $C>1$, for large enough $n$.

— Density conditions for $k$ vertex-disjoint triangles in tripartite graphs  (2503.05218 - Guo et al., 7 Mar 2025) in Conjecture 1, immediately after Remark following Theorem 1