On a conjecture of Manickam and Singhi

Let In = {1; 2; : : : ; n} and x : In → R be a map such that ∑i∈In xi ¿ 0. (For any i, its image is denoted by xi.) Let F = {J ⊂ In : |J | = k, and ∑j∈J xj ¿ 0}. Manickam and Singhi (J. Combin. Theory Ser. A 48 (1) (1988) 91–103) have conjectured that |F|¿ ( n−1 k−1 ) whenever n¿ 4k and showed that the conclusion of the conjecture holds when k divides n. For any two integers r and ‘ let [r]‘ denote the smallest positive integer congruent to r (mod ‘). 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− 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¿ 2ek. c © 2003 Elsevier B.V. All rights reserved.