Completely separating system(CSS) which is also an antichain (Sperner family)
Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.
Is this studied in the literature?
I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)
combinatorics order-theory
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
add a comment |
Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.
Is this studied in the literature?
I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)
combinatorics order-theory
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32
add a comment |
Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.
Is this studied in the literature?
I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)
combinatorics order-theory
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.
Is this studied in the literature?
I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)
combinatorics order-theory
combinatorics order-theory
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
edited Nov 28 '18 at 7:33
Cyriac Antony
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
asked Nov 28 '18 at 5:32
Cyriac AntonyCyriac Antony
12910
12910
Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32
add a comment |
Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32
Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32
Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32
add a comment |
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Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32