On a conjecture of Manickam and Singhi

Let In={1,2,…,n} and x : In↦R be a map such that ∑i∈Inxi⩾0. (For any i, its image is denoted by xi.) Let F={J⊂In:|J|=k, and ∑j∈Jxj⩾0}. Manickam and Singhi (J. Combin. Theory Ser. A 48 (1) (1988) 91–103) have conjectured that |F|⩾(n−1k−1) whenever n⩾4k and showed that the conclusion of the conjecture holds when k divides n. For any two integers r and ℓ let [r]ℓ denote the smallest positive integer congruent to r (mod ℓ). Bier and Manickam (Southeast Asian Bull. Math. 11 (1) (1987) 61–67) have shown that if k>3 and n⩾k(k−1)k(k−2)k+k(k−1)2(k−2)+k[n]k then the conjecture holds. In this note, we give a short proof to show that the conjecture holds when n⩾2k+1ekkk+1.