The Erdos-Jacobson-Lehel conjecture on potentially Pk-graphic sequence is true

A variation in the classical Turan extremal problem is studied. A simple graph G of order n is said to have property P-k if it contains a clique of size k + 1 as its subgraph. An n-term nonincreasing nonnegative integer sequence pi = (d(1), d(2),..., d(n)) is said to be graphic if it is the degree sequence of a simple graph G of order n and such a graph G is referred to as a realization of;(. A graphic sequence a is said to be potentially PI-graphic if it has a realization G having property P-k. The problem: determine the smallest positive even number sigma(k, n) such that every n-term graphic sequence pi =(d(1), d(2),..., d(n)) without zero terms and with degree sum sigma(pi) = d(1) + d(2) + ... + d(n) at least sigma(k, n) is potentially P-k-graphic has been proved positive.