First distribution invariants and EKR theorems

A linear programming approach to the Manickam–Miklós–Singhi conjecture

“…The following sums generated by PropagateNegative (Algorithm 2) must be strictly negative or we have at least 286 nonnegative sets: 14 x 1 + x 8 + x 11 + x 14 x 2 + x 4 + x 11 + x 14 x 2 + x 5 + x 9 + x 14 x 2 + x 5 + x 11 + x 13 x 2 + x 6 + x 8 + x 14 x 2 + x 6 + x 9 + x 13 x 2 + x 7 + x 10 + x 12 x 3 + x 4 + x 10 + x 14 x 3 + x 4 + x 12 + x 13 x 3 + x 5 + x 8 + x 14 x 3 + x 5 + x 9 + x 13 x 3 + x 5 + x 10 + x 12 x 3 + x 6 + x 7 + x 14 x 3 + x 6 + x 8 + x 13 x 3 + x 6 + x 9 + x 12 x 3 + x 7 + x 8 + x 12 x 3 + x 7 + x 10 + x 11 x 3 + x 8 + x 9 + x 11 x 4 + x 5 + x 7 + x 14 x 4 + x 5 + x 8 + x 13 x 4 + x 5 + x 9 + x 12 x 4 + x 6 + x 7 + x 13 x 4 + x 6 + x 8 + x 12 x 4 + x 6 + x 9 + x 11 x 5 + x 7 + x 8 + x 11 .…”

“…The following sums generated by PropagateNegative (Algorithm 2) must be strictly negative or we have 364 nonnegative sets: 15 x 1 + x 5 + x 9 + x 14 x 1 + x 5 + x 10 + x 13 x 1 + x 6 + x 8 + x 14 x 1 + x 6 + x 9 + x 13 x 1 + x 7 + x 11 + x 12 13 x 2 + x 5 + x 9 + x 12 x 2 + x 6 + x 8 + x 12 x 2 + x 6 + x 10 + x 11 x 2 + x 7 + x 9 + x 11…”

“…Bier and Manickam [4] showed that every nonnegative sum  n i=1 x i has at least 2 n−1 nonnegative partial sums. Manickam, Miklós, and Singhi [12,13] considered the situation when each partial sum has exactly k terms (we call such partial sums k-sums), and they conjectured there are many nonnegative k-sums when n is sufficiently large. [12,13]  is sharp since the example x 1 = n − 1 and x i = −1 for i ̸ = 1 has sum zero and a k-sum is nonnegative exactly when it contains x 1 .…”

A linear bound on the Manickam–Miklós–Singhi conjecture

“…In [16], Conjecture 1.1 was phrased in the combinatorial form in which it is stated above. In this paper we will speak only about the combinatorial version-we refer the reader to [17,3] for more details about the association scheme version.…”

“…Manickam, Miklós, and Singhi conjectured that for n ≥ 4k this assignment gives the least possible number of nonnegative k-sums. Manickam, Miklós, Singhi [16,17].) Suppose that n ≥ 4k, and we have n real numbers x 1 , .…”