  We investigate the number of 4-edge paths in graphs with a fixed number of
vertices and edges. An asymptotically sharp upper bound is given to this
quantity. The extremal construction is the quasi-star or the quasi-clique
graph, depending on the edge density. An easy lower bound is also proved. This
answer resembles the classic theorem of Ahlswede and Katona about the maximal
number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun
about k-edge stars.
