### Updates to the Manickam-Miklós-Singhi Conjecture

#### by Derrick Stolee

It has been a busy year for the Manickam-Miklós-Singhi Conjecture. (See the earlier post on this topic for the important definitions.) Several new papers have appeared online or on the arXiv. Here is a brief discussion of these recent developments (organized by time of submission, as far as I can tell).

Recall that for , the function stores the minimum integer such that for all every vector with nonnegative total sum has at least nonnegative -sums. The conjecture states that .

M. Tyomkyn, An improved bound for the Manickam-Miklós-Singhi Conjecture. *European Journal of Combinatorics* 33 (2012) 27-32.

Tyomkyn improves previous exponential bounds to find .

N. Alon, H. Huang, and B. Sudakov. Nonnegative -sums, fractional covers, and probability of small deviations. *Journal of Combinatorial Theory, Series B*, 103(3) (2012) 784-796.

Using the idea of fractional covers, the authors prove the first polynomial upper bound of .

A. Chowdhury, A note on the Manickam-Miklós-Singhi Conjecture, *European Journal of Combinatorics*, Volume 35, January 2014, Pages 131–140.

Chowdhury extends her thesis work to provide new upper bounds of , , and . The technique is a computer-assisted proof by using the computer to generate a certain type of cover on a hypergraph. This hypergraph contains sets that **must** be negative in order to avoid having at least nonnegative -sums. The way these sets are classified as such uses the shift poset and an interesting application of the Kruskal-Katona Theorem.

S. G. Hartke, D. Stolee, A linear programming approach to the Manickam-Miklós-Singhi Conjecture, to appear in *European Journal of Combinatorics*.

This is the work described in the previous post. Using a linear programming formulation and a very involved computer search, the authors verify that , , , and . Also, the extremal examples for small and are demonstrated and this provides a conjecture that limits to a value approximately 3.14157899.

P. Frankl, On the number of nonnegative sums, to appear in *Journal of Combinatorial Theory, Series B*.

Frankl produces a slightly weaker but **much** shorter proof than Alon, Huang, and Sudakov, showing . Note this is a stronger result than Alon, Huang, and Sudakov’s cubic bound.

A. Pokrovskiy, A linear bound on the Manickam-Mikl ́ós-Singhi Conjecture, arXiv:1308.2176v1.

Pokrovskiy proves a linear bound of , although the constant term is . This is a very nice improvement, and I look forward to people lowering this constant through more careful analysis of his proof. However, Pokrovskiy points out that his proof as written will not achieve a value of .

Have a new update? Working on your own results? Send me an email so I can update this page.

I am now working on Manickam-Miklos-Singhi conjecture for projective spaces.

Manickam

Tyomkyn’s bound was (logk)^k.

Misha Tyomkyn.

Thanks for catching my mistake.

I am glad to note that there is so much interest shown by mathematicians all around the world. N. Manickam, Chair, Dept of Math, Depauw University, Greencastle, In 46135 USA. E-mail:manickam@depauw.edu