When analyzing one poset graph, can an element be both in the longest chain and the longest antichain? (or do chains and antichains have to be mutually exclusive?)
2026-03-27 04:18:38.1774585118
Element in chain and antichain
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An element could be both in a longest chain and a longest anti-chain. Consider the boolean lattice (subsets of an $n$-element set). Any full chain (from the top to the bottom) is a longest chain. The middle row is a longest anti-chain. But any full chain will hit an element in the middle row.