# Choice Functions

@inproceedings{Aharoni2021ChoiceF, title={Choice Functions}, author={Ron Aharoni and Joseph Briggs}, year={2021} }

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

#### One Citation

On two parametric probability distributions on crisp complete pre-orders

- Biology
- 2019

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

#### References

SHOWING 1-10 OF 53 REFERENCES

Transversals in Row-Latin Rectangles

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 1998

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 the… Expand

A Weak Version of Rota's Bases Conjecture for Odd Dimensions

- Mathematics, Computer Science
- SIAM J. Discret. Math.
- 2014

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

- Mathematics
- 1935

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 of… Expand

The intersection of a matroid and a simplicial complex

- Mathematics
- 2006

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 one… Expand

Domination numbers and homology

- 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

- Mathematics, Computer Science
- Comb.
- 2019

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

- Mathematics, Computer Science
- J. Graph Theory
- 2018

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

- Computer Science, Mathematics
- SIAM J. Discret. Math.
- 2004

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

- Mathematics, Computer Science
- SIAM J. Discret. Math.
- 2018

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.

- Mathematics
- 2020

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 there… Expand