Sperners theorem is about antichains (subsets of powerset of n elements for which no pair of elements contains the other) for example if we choose from
row 5: {a,b,c,d} row 4: {a,b,c} {a,b,d} {a,c,d} {b,c,d} row 3: {a,b} {a,c} {a,d} {b,c} {b,d} {c,d} <-- middle row is biggest row 2: {a} {b} {c} {d} row 1: {}
we find {a,b,c} {a,c,d} {b,d}
is an antichain of size 3. It looks like the biggest antichain is the middle row {a,b} {a,c} {a,d} {b,c} {b,d} {c,d}
.
To prove that is always the case take an antichain and pick all the elements from the lowest row R below the middle (flip the antichain upside down by taking the complement of every element if it's not below the middle). There must be at least |R| gaps in the above row: so push everything in the bottom row up one and it's still an antichain. Repeat until everything is in the middle row.
Any ideas where the mistake in this is?