Compute the four-dimensional maximal absolute projection constant λ(4)

Determine the exact value of the real maximal absolute projection constant λ(4), that is, the supremum over N ≥ 4 of the maximal norm of a projection from ℓ∞^N onto a 4-dimensional subspace.

Background

The paper proves that for every k ≥ 2 and every graph G on n ≥ k vertices, the upper bound on the k-th adjacency eigenvalue λk(G) is controlled by the real maximal absolute projection constant λ(k−1).

For k = 5, this reduces the best possible bound on c5 to knowing λ(4). The authors note the current best upper bound λ(4) ≤ (2 + 3√6)/5 and that determining λ(4) would sharpen or settle the corresponding extremal eigenvalue problem for λ5(G).

References

Thus the method solves the $\lambda_4$ problem exactly and reduces the {upper bound} on the $\lambda_5$ problem to the open problem of computing the four dimensional projection constant \lambda_{}(4).

— Graph Eigenvalues and Projection Constants  (2603.29280 - Wakhare, 31 Mar 2026) in Introduction

We conjecture that the codimension assumption can be removed and that the same estimate holds for every subspace $V\subset\ell_\inftyN$ with $\lambda(V)<4/3$.

— Projection Constants and Banach-Mazur Distance in Codimension Two  (2610.06141 - Deregowska et al., 5 Oct 2026) in Abstract; Conjecture 1 in Section 1 (labelled \ref{coniektore})