Stability of associated primes of integral closures of monomial ideals

Let I be a monomial ideal of a polynomial ring R = K [ X 1 , … , X r ] and d ( I ) the maximal degree of minimal generators of I. In this paper, we explicitly determine a number n 0 in terms of r and d ( I ) such that Ass ( R / I n ¯ ) = Ass ( R / I n 0 ¯ ) for all n ⩾ n 0 . Furthermore, our n 0 is almost sharp.