Corpus ID: 236447539

Choice Functions

  title={Choice Functions},
  author={Ron Aharoni and Joseph Briggs},
This is a survey paper on rainbow sets (another name for “choice functions”). The main theme is the distinction between two types of choice functions: those having a large (in the sense of belonging to some specified filter, namely closed up set of sets) image, and those that have a large domain and small image, where “smallness” means belonging to some specified complex (a closed-down set). The paper contains some new results: (1) theorems on scrambled versions, in which the sets are re… Expand
On two parametric probability distributions on crisp complete pre-orders
This paper determines a parametric probability distribution on pre- orders generalizing Mallows Distribution by considering pre-orders as orders on blocks of equivalent elements and generalizing the Plackett-Luce distribution on complete pre- Orders. Expand


Transversals in Row-Latin Rectangles
It is shown that anm×nrow-latin rectangle with symbols in {1,2,?,k},k?n, has a transversal wheneverm?2n?1, and that this lower bound formis sharp. Several applications are given. One is theExpand
A Weak Version of Rota's Bases Conjecture for Odd Dimensions
The Alon--Tarsi Latin squares conjecture is extended to odd dimensions by stating it for reduced Latin squares (Latin squares having the identity permutation as their first row and first column) and it is shown that the validity of this conjecture implies a weak version of Rota's bases conjecture for odd dimensions. Expand
On Representatives of Subsets
Let a set S of mn things be divided into m classes of n things each in two distinct ways, (a) and (b); so that there are m (a)-classes and m (b)-classes. Then it is always possible to find a set R ofExpand
The intersection of a matroid and a simplicial complex
A classical theorem of Edmonds provides a min-max formula relating the maximal size of a set in the intersection of two matroids to a "covering" parameter. We generalize this theorem, replacing oneExpand
Domination numbers and homology
  • R. Meshulam
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 2003
The following Hall-type conjecture of Aharoni is proved: If γs*(G) denotes the fractional star-domination number of G and let V =∪i=1m Vi be a partition of V into m classes then G contains an independent set which intersects all m classes. Expand
Rainbow Fractional Matchings
It is proved that any family of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$ has a rainbow fractional matches of size £n. Expand
Degree Conditions for Matchability in 3-Partite Hypergraphs
A strong version of a theorem of Drisko (as generalized by the first two authors) is proved, that every family of matchings of size $2n-1$ in a bipartite graph has a partial rainbow matching of size n. Expand
Finding Large Independent Sets in Graphs and Hypergraphs
Here, it is shown that an RNC algorithm due to Beame and Luby finds an independent set of expected size $\alpha_k(H)$ and also derandomizes it for certain special cases. Expand
Rainbow matchings in properly-coloured multigraphs
It is shown that in any bipartite multigraph that is properly edge-coloured by colours with at least $n + o(n)$ edges of each colour there must be a matching of size $n-O(1)$ that uses each colour at most once. Expand
Cooperative conditions for the existence of rainbow matchings.
Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets of edges in a bipartite graph. If the union of every $k$ members of $\mathcal{F}$ contains a matching of size $n$, then thereExpand