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It is known that given an infinite poset, it always contains an infinite chain or antichain; moreover, there is a constructive proof that we can find a continuous chain in $P(\mathbb{N})$; so, in general, I'm asking if given a poset of a certain cardinality, we could always find a chain or antichain of the same cardinality.

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Maybe off topic. What is a "continuous chain"? – Srivatsan Sep 14 '11 at 0:47
@Srivatsan, in this case it seems to mean a chain of cardinality $2^{\aleph_0}$. – Henning Makholm Sep 14 '11 at 1:24
@Srivatsan, yes Henning is right. – Immanuel Weihnacht Sep 14 '11 at 10:07
up vote 10 down vote accepted

In fact, there is a negative answer provable in ZFC. Since we have choice, well-order the interval $[0,1]$. Then let $x\sqsubset y$ if and only if the well-order agrees with the standard order of the reals and $x<y$. Then $[0,1]$ with the ordering $\sqsubset$ is an uncountable poset with neither an uncountable chain nor an uncountable anti-chain.

Consider any uncountable $S\subseteq[0,1]$. Let $z$ be the infimum of the set $$\{x\in [0,1]|\text{ there are uncountably many }y<x\text{ with }y\in S\}$$ $S$ cannot be a chain since otherwise, a countable increasing sequence of elements of $S$ converging to $z$ would be cofinal in $\omega_1$.

Similarly, we can take $w$ to be the supremum of the set $$\{x\in [0,1]|\text{ there are uncountably many }y>x\text{ with }y\in S\}$$ Then if $S$ was an anti-chain, then a countable decreasing sequence (according to the standard ordering of the reals) in $S$ would again be cofinal in $\omega_1$.

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I don't understand your definition of $x \sqsubset y$: what are $x$ and $y$ here and how do they figure into the rest of the definition? – Pete L. Clark Sep 14 '11 at 1:09
Sorry, x and y are taken to be real numbers in [0,1]. – Aubrey da Cunha Sep 14 '11 at 1:10
To be clear, in the second and third paragraphs, do all references to order (infimum, $<$, increasing, etc) refer to the order defined in the first paragraph? – Jason DeVito Sep 14 '11 at 1:10
No, they refer to the standard ordering of the reals. – Aubrey da Cunha Sep 14 '11 at 1:15
It's perhaps worth mentioning that this clever argument is attributed to Sierpinski. – user83827 Sep 14 '11 at 2:55

A Suslin tree is a poset (in fact, a tree) that has cardinality $\omega_1$ but every chain and every antichain is countable. The existence of a Suslin tree is neither provable nor disprovable in ZFC. Therefore, your question does not have an affirmitive answer provable in ZFC.

I do not know whether it has a negative answer (using something other than Suslin trees) provable in ZFC. Aubrey da Cunha has given a proof that the answer to the question is "no". That answer should be accepted over this one.

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A natural negative answer is to take the poset $\bigcup_{n\in\omega} \{n\}\times\omega_n$, with two points comparable iff their first coordinates coincide. Any antichain here is countable, and any chain has size strictly below $\aleph_\omega$, which is the size of the whole set. Note that this does not use any form of choice.

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This is true exactly for $\omega$ and for weakly compact cardinals. It is proved in

G.D. Badenhorst and T. Sturm, An order- and graph-theoretical characterisation of weakly compact cardinals, in Cycles and Rays (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 301), 1990, pp.19-20, MR1096981.

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