The size of 3-compatible, weakly compatible split systems
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A split system on a finite set X is a set of bipartitions of X. Weakly compatible and k-compatible (k≥1) split systems are split systems which satisfy special restrictions on all subsets of a certain fixed size. They arise in various areas of applied mathematics such as phylogenetics and multi-commodity flow theory. In this note, we show that the number of splits in a 3-compatible, weakly compatible split system on a set X of size n is linear in n.
KeywordsPhylogenetic combinatorics Extremal combinatorics of finite sets Split systems Compatibility Weak compatibility
Mathematics Subject Classification (2000)05D05 03E05 92B05
TW and VM were supported by the Engineering and Physical Sciences Research Council [grant number EP/D068800/1]. VM thanks the Royal Society for enabling him to visit TW and JK in Singapore. TW was also partially supported by the Singapore MOE grant R-146-000-134-112. JK was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number 2010-0008138).
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