Related papers: Choice functions in the intersection of matroids
A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on…
A {\em connectivity function} on a set $E$ is a function $\lambda:2^E\rightarrow \mathbb R$ such that $\lambda(\emptyset)=0$, that $\lambda(X)=\lambda(E-X)$ for all $X\subseteq E$, and that $\lambda(X\cap Y)+\lambda(X\cup Y)\leq…
Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…
We associate to certain symmetric or antisymmetric functions on the set ${E\choose d+1}$ of $(d+1)-$subsets in a finite set $E$ an equivalence relation on $E$ and study some of its properties.
An exact r-coloring of a set $S$ is a surjective function $c:S \rightarrow \{1, 2, \ldots,r\}$. A rainbow solution to an equation over $S$ is a solution such that all components are a different color. We prove that every 3-coloring of…
We introduce neutrosophic choice functions, the neutrosophic counterpart of the Axiom of Choice, prove some results, and discuss how it effects the foundations of mathematics in a neutrosophic setting.
Suppose we are given a family of choice functions on pairs from a given finite set (with at least three elements) closed under permutations of the given set. The set is considered the set of alternatives (say candidates for an office). The…
Polynomial functions $f : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ are studied for which sums of arbitrary length $f (1) + f (2) + f (3) + >... + f (n)$, with $n \in \mathbb{N}_+$, can be expressed by polynomial functions $g :…
The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n>=1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case…
Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented…
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough…
We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of…
Suppose we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential "voters". The question is, what choice…
In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem…
We prove the following colorful Helly-type result: Fix $k \in [d-1]$. Assume $\mathcal{A}_1, \dots, \mathcal{A}_{d+(d-k)+1}$ are finite sets (colors) of nonzero vectors in $\R^d$. If for every rainbow sub-selection $R$ from these sets of…
For each proper minor-closed subclass $\cM$ of the $\GF(q^2)$-representable matroids containing all simple $\GF(q)$-representable matroids, we give, for all large $r$, a tight upper bound on the number of points in a rank-$r$ matroid in…
Given a graph $H$, let $g(n,H)$ denote the smallest $k$ for which the following holds. We can assign a $k$-colouring $f_v$ of the edge set of $K_n$ to each vertex $v$ in $K_n$ with the property that for any copy $T$ of $H$ in $K_n$, there…
A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of…
We study projective functions. We prove that projective functions generalise lower and upper-semianalytic ones while being stable by composition and difference. We show that the class of projective functions is closed under sums,…
Difference balanced functions from $F_{q^n}^*$ to $F_q$ are closely related to combinatorial designs and naturally define $p$-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are…