Abstract The linear projection constant Π ( E ) of a finite-dimensional real Banach space E is the smallest number C ∈ [ 0 , + ∞ ) such that E is a C -absolute retract in the category of real Banach spaces with bounded linear maps. We denote by Π n the maximal linear projection constant amongst n -dimensional Banach spaces. In this article, we prove that Π n may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the maximal relative projection constants of codimension n converge to 1 + Π n . Furthermore, using the classification of K 4 -free two-graphs, we give an alternative proof of Π 2 = 4 3 . We also show by means of elementary functional analysis that for each integer n ⩾ 1 there exists a polyhedral n -dimensional Banach space F n such that Π ( F n ) = Π n .

The maximal linear projection constant n can be determined by computing eigenvalues of certain two-graphs, and for each integer n  1, there exists a polyhedral n-dimensional Banach space Fn such that (Fn) = n.

Computation of maximal projection constants

Key Takeaway: The maximal linear projection constant n can be determined by computing eigenvalues of certain two-graphs, and for each integer n 1, there exists a polyhedral n-dimensional Banach space Fn such that (Fn) = n.