The Erdös-Jacobson-Lehel conjecture on potentiallyPk-graphic sequence is true

A variation in the classical Turan extrernal problem is studied. A simple graphG of ordern is said to have propertyPk if it contains a clique of sizek+1 as its subgraph. Ann-term nonincreasing nonnegative integer sequence π=(d1, d2,⋯, d2) is said to be graphic if it is the degree sequence of a simple graphG of ordern and such a graphG is referred to as a realization of π. A graphic sequence π is said to be potentiallyPk-graphic if it has a realizationG having propertyPk. The problem: determine the smallest positive even number σ(k, n) such that everyn-term graphic sequence π=(d1, d2,…, d2) without zero terms and with degree sum σ(π)=(d1+d2+ …+d2) at least σ(k,n) is potentially Pk-graphic has been proved positive.