  In this paper, we develop a virtual element method (VEM) of high order to
solve the fourth order plate buckling eigenvalue problem on polygonal meshes.
We write a variational formulation based on the Kirchhoff-Love model depending
on the transverse displacement of the plate. We propose a $C^1$ conforming
virtual element discretization of arbitrary order $k\ge2$ and we use the
so-called Babuska--Osborn abstract spectral approximation theory to show that
the resulting scheme provides a correct approximation of the spectrum and prove
optimal order error estimates for the buckling modes (eigenfunctions) and a
double order for the buckling coefficients (eigenvalues). Finally, we report
some numerical experiments illustrating the behaviour of the proposed scheme
and confirming our theoretical results on different families of meshes.
