The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length

Let v ( k ) be the smallest integer larger than 1 that does not occur among the denominators in any identity of the form 1 = 1 n 1 + ⋯ + 1 n k , {align*}1=1{n_1}+⋯+1{n_k},{align*} where 1 less than or equals n 1 less than midline horizontal ellipsis less than n Subscript k 1 ≤ n 1 < ⋯ < n k 1 ≤ n₁<⋯< nₖ are pairwise distinct integers. In their 1980 monograph, Erdős and Graham asked for quantitative estimates on the growth of v ( k ) and suggested the lower bound v left parenthesis k right parenthesis much greater than k factorial v ( k ) ≫ k ! v(k) k! . In this paper we give the first known improvement and show that there exists an absolute constant c greater than 0 c > 0 c>0 such that the inequality v ( k ) ≥ e c k 2 {align*}v(k)≥ ec k^2{align*} holds for all positive integers k .