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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 scrambl
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We prove a common generalization of two results, one on rainbow fractional matchings and one on rainbow sets in the intersection of two matroids: Given $d = r lceil k rceil - r + 1$ functions of size (=sum of values) $k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i)$, $i leq d$, supporting a function with the same properties.
A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A graph $G$ is strongly fractional $r$-choosable if $G$ is $(a,b)$-choosable for all positive integers $a,b$ for which $a/b ge r$. The strong fractional choice number of $G$ is $ch_f^s(G) = inf {r: G $ is strongly fractional $r$-choosable$}$. This paper determines the strong fractional choice number of all $3$-choice critical graphs.
A social choice correspondence satisfies balancedness if, for every pair of alternatives, x and y, and every pair of individuals, i and j, whenever a profile has x adjacent to but just above y for individual i while individual j has y adjacent to but just above x, then only switching x and y in the orderings for both of those two individuals leaves the choice set unchanged. We show how the balancedness condition interacts with other social choice properties, especially tops-only. We also use balancedness to characterize the Borda rule (for a fixed number of voters) within the class of scoring rules.
In this short note, we show that for any $epsilon >0$ and $k<n^{0.5-epsilon}$ the choice number of the Kneser graph $KG_{n,k}$ is $Theta (nlog n)$.
We study conditions for the existence of stable and group-strategy-proof mechanisms in a many-to-one matching model with contracts if students preferences are monotone in contract terms. We show that equivalence, properly defined, to a choice profile under which contracts are substitutes and the law of aggregate holds is a necessary and sufficient condition for the existence of a stable and group-strategy-proof mechanism. Our result can be interpreted as a (weak) embedding result for choice functions under which contracts are observable substitutes and the observable law of aggregate demand holds.
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