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).