Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach

In 2011, Neil Hindman proved that for every natural number $n,m$ the polynomial \begin{equation*} \sum_{i=1}^{n} x_{i}-\prod\limits_{j=1}^{m} y_{j} \end{equation*} has monochromatic solutions for every finite coloration of $\mathbb{N}$. We want to generalize this result to two classes of nonlinear polynomials. The first class consists of polynomials $P(x_{1},...,x_{n},y_{1},...,y_{m})$ of the following kind: \begin{equation*} P(x_{1},...,x_{n},y_{1},...,y_{m})=\sum_{i=1}^{n}a_{i}x_{i}M_{i}(y_{1},...,y_{m}), \end{equation*} where $n,m$ are natural numbers, $\sum\limits_{i=1}^{n}a_{i}x_{i}$ has monochromatic solutions for every finite coloration of $\mathbb{N}$ and the degree of each variable $y_{1},...,y_{m}$ in $M_{i}(y_{1},...,y_{m})$ is at most one. An example of such a polynomial is \begin{equation*} x_{1}y_{1}+x_{2}y_{1}y_{2}-x_{3}.\end{equation*} The second class of polynomials generalizing Hindman's result is more complicated to describe; its particularity is that the degree of some of the involved variables can be greater than one.\\ The technique that we use relies on an approach to ultrafilters based on Nonstandard Analysis. Perhaps, the most interesting aspect of this technique is that, by carefully chosing the appropriate nonstandard setting, the proof of the main results can be obtained by very simple algebraic considerations

Nonnegative k-sums, fractional covers, and probability of small deviations

More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for any integers $n, k$ satisfying $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for $n \geq 33k^2$. This substantially improves the best previously known exponential lower bound $n \geq e^{ck \log\log k}$. In addition we prove a tight stability result showing that for every $k$ and all sufficiently large $n$, every set of $n$ reals with a nonnegative sum that does not contain a member whose sum with any other $k-1$ members is nonnegative, contains at least $\binom{n-1}{k-1}+\binom{n-k-1}{k-1}-1$ subsets of cardinality $k$ with nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde