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Combinatorics » Posets

Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

Question   Does there exist a constant $ c>1/2 $ and a function $ n_0(k) $ such that if $ |X|\geq n_0(k) $, then every saturated $ k $-Sperner system $ \mathcal{F}\subseteq \mathcal{P}(X) $ has cardinality at least $ 2^{(1+o(1))ck} $?

Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system

Posted by Jon Noel
updated April 12th, 2014
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Graph Theory » Coloring » Nowhere-zero flows

Antichains in the cycle continuous order ★★

Author(s): DeVos

If $ G $,$ H $ are graphs, a function $ f : E(G) \rightarrow E(H) $ is called cycle-continuous if the pre-image of every element of the (binary) cycle space of $ H $ is a member of the cycle space of $ G $.

Problem   Does there exist an infinite set of graphs $ \{G_1,G_2,\ldots \} $ so that there is no cycle continuous mapping between $ G_i $ and $ G_j $ whenever $ i \neq j $ ?

Keywords: antichain; cycle; poset

Posted by mdevos
updated May 12th, 2007
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