Erdős four-edge intersection equality

Determine whether the minimum-intersection function for four common edges satisfies f(n,4)=2n-4 for all n≥6, as proposed using the complete bipartite graph K_{2,n-2}.

Background

For an n-vertex graph G, define f(n,4) as the minimum number of edges such that every permutation of the vertex set produces a relabelled graph sharing at least four edges with G. Erdős established the upper bound f(n,4)≤2n−4, witnessed by K_{2,n−2}, and posed equality as the next unresolved case after the known values for k=1,2,3.

The paper proves only the asymptotic relation f(n,4)=2n−o(n), together with the bounds 2n−10n{2/3}−7<f(n,4)≤2n−4. Thus the exact equality f(n,4)=2n−4 remains unresolved by the results presented.

References

Erdős then considered the small values of k. He recorded f(n,1)=n-1 and, citing Chung, Graham, and Murty, f(n,2)=f(n,3)=\left\lfloor\frac{3n}{2}\right\rfloor, and left the next two cases, k=4 and k=5, open. For k=4 he observed \begin{equation}\label{upper-four} f(n,4)\leq 2n-4\qquad(n\geq6), \end{equation} using the complete bipartite graph K_{2,n-2}, and asked whether \begin{equation}\label{erdos-conjecture} f(n,4)=2n-4. \end{equation}

upper-four:

f(n,4)≤2n−4(n≥6),f(n,4)\leq 2n-4\qquad(n\geq6),

erdos-conjecture:

f(n,4)=2n−4.f(n,4)=2n-4.

— An asymptotic solution to the Erdős four-edge intersection problem  (2608.17727 - Żak, 18 Aug 2026) in Section 1, Introduction