相关论文: The Manickam-Mikl\'os-Singhi Conjectures for Sets …

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. Define a real-valued weight function on the $1$-dimensional vector spaces of $V$ such that the sum of all weights is zero. Let the weight of a subspace $S$ be…

More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is…

We show that for $n>k(4e\log k)^k$ every set $\{x_1,..., x_n\}$ of $n$ real numbers with $\sum_{i=0}^{n}x_i \geq 0$ has at least $\binom{n-1}{k-1}$ $k$-element subsets of a non-negative sum. This is a substantial improvement on the best…

Suppose that we have a set of numbers x_1, ..., x_n which have nonnegative sum. How many subsets of k numbers from {x_1, ..., x_n} must have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n at least 4k the answer is (n-1…

In 1998 Manickam and Singhi conjectured that for every positive integer $d$ and every $n \ge 4d$, every set of $n$ real numbers whose sum is nonnegative contains at least $\binom {n-1}{d-1}$ subsets of size $d$ whose sums are nonnegative.…

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…

An r-unform n-vertex hypergraph H is said to have the Manickam-Mikl\'os-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the…

The Manickam-Miklos-Singhi Conjecture states that when n is at least 4k, every multiset of n real numbers with nonnegative total sum has at least (n-1 choose k-1) k-subsets with nonnegative sum. We develop a branch-and-cut strategy using a…

The Erd\H{o}s-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any…

A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: For each pair of positive integers $(n,k)$ with $3\le k\le n-3$, there is a $k$-dimensional $\Bbb F_2$-subspace $E$…

M\'oricz and Nagy introduced the problem of maximizing the number of $r$-element subsets with rational sums in an $n$-element set of irrational numbers, and showed that it is equivalent to an extremal zero-sum problem. They determined the…

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

The well-known Lvov-Kaplansky conjecture states that the image of a multilinear polynomial $f$ evaluated on $n\times n$ matrices is a vector space. A weaker version of this conjecture, known as the Mesyan conjecture, states that if $m=deg(…

Twenty years ago Bondy and Vince conjectured that for any nonnegative integer $k$, except finitely many counterexamples, every graph with $k$ vertices of degree less than three contains two cycles whose lengths differ by one or two. The…

In 1961, P. Erd\H{o}s, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n$. We will prove a similar result for pairs of…

For a compact set $A \subset {\mathbb R}^d$ and an integer $k\ge 1$, let us denote by $$ A[k] = \left\{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right\}=\sum_{i=1}^k A$$ the Minkowski sum of $k$ copies of $A$. A theorem of Shapley, Folkmann…

In 1933 Karol Borsuk asked whether each bounded set in the n-dimensional Euclidean space can be divided into n+1 parts of smaller diameter. The diameter of a set is defined as the supremum (least upper bound) of the distances of contained…

In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\H{o}s conjectured in 1990 that the number of such…

Let $X_0^{\star}(k,n,s)$ denote the sum of all multiple zeta-star values of weight $k$, depth $n$ and height $s$. Kaneko and Ohno conjecture that for any positive integers $m,n,s$ with $m,n\geqslant s$, the difference…