Exploring Agreeability in Tree Societies
Let S be a collection of convex sets in Rd with the property that any subcollection of d − 1 sets has a nonempty intersection. Helly’s Theorem states that ∩s∈S S is nonempty. In a forthcoming paper, Berg et al. (Forthcoming) interpret the one dimensional version of Helly’s Theorem in the context of...
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ndltd-CLAREMONT-oai-scholarship.claremont.edu-hmc_theses-12212019-10-16T03:06:13Z Exploring Agreeability in Tree Societies Fletcher, Sarah Let S be a collection of convex sets in Rd with the property that any subcollection of d − 1 sets has a nonempty intersection. Helly’s Theorem states that ∩s∈S S is nonempty. In a forthcoming paper, Berg et al. (Forthcoming) interpret the one dimensional version of Helly’s Theorem in the context of voting in a society. They look at the effect that different intersection properties have on the proportion of a society that must agree on some point or issue. In general, we define a society as some underlying space X and a collection S of convex sets on the space. A society is (k, m)-agreeable if every m-element subset of S has a k-element subset with a nonempty intersection. The agreement number of a society is the size of the largest subset of S with a nonempty intersection. In my work I focus on the case where X is a tree and the convex sets in S are subtrees. I have developed a reduction method that makes these tree societies more tractable. In particular, I have used this method to show that the agreement number of (2, m)-agreeable tree societies is at least 1 |S | and 3 that the agreement number of (k, k + 1)-agreeable tree societies is at least |S|−1. 2009-05-01T07:00:00Z text application/pdf https://scholarship.claremont.edu/hmc_theses/218 https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1221&context=hmc_theses HMC Senior Theses Scholarship @ Claremont Tree Societies Helly’s Theorem |
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Tree Societies Helly’s Theorem Fletcher, Sarah Exploring Agreeability in Tree Societies |
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Let S be a collection of convex sets in Rd with the property that any subcollection of d − 1 sets has a nonempty intersection. Helly’s Theorem states that ∩s∈S S is nonempty. In a forthcoming paper, Berg et al. (Forthcoming) interpret the one dimensional version of Helly’s Theorem in the context of voting in a society. They look at the effect that different intersection properties have on the proportion of a society that must agree on some point or issue. In general, we define a society as some underlying space X and a collection S of convex sets on the space. A society is (k, m)-agreeable if every m-element subset of S has a k-element subset with a nonempty intersection. The agreement number of a society is the size of the largest subset of S with a nonempty intersection. In my work I focus on the case where X is a tree and the convex sets in S are subtrees. I have developed a reduction method that makes these tree societies more tractable. In particular, I have used this method to show that the agreement number of (2, m)-agreeable tree societies is at least 1 |S | and 3 that the agreement number of (k, k + 1)-agreeable tree societies is at least |S|−1. |
author |
Fletcher, Sarah |
author_facet |
Fletcher, Sarah |
author_sort |
Fletcher, Sarah |
title |
Exploring Agreeability in Tree Societies |
title_short |
Exploring Agreeability in Tree Societies |
title_full |
Exploring Agreeability in Tree Societies |
title_fullStr |
Exploring Agreeability in Tree Societies |
title_full_unstemmed |
Exploring Agreeability in Tree Societies |
title_sort |
exploring agreeability in tree societies |
publisher |
Scholarship @ Claremont |
publishDate |
2009 |
url |
https://scholarship.claremont.edu/hmc_theses/218 https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1221&context=hmc_theses |
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AT fletchersarah exploringagreeabilityintreesocieties |
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