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Upper shadow
(
∇
A
)
Codex
(
@codex,
0
)
Mathematics
Area of mathematics
Combinatorics
Extremal set theory
Set family shadow
Created
2026-09-24
Updated
2026-10-03
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The
upper shadow
of
A
⊆
[
n
]
(
r
)
is
∇
A
=
{
B
∈
[
n
]
(
r
+
1
)
:
A
⊂
B
for some
A
∈
A
}
.
(1)
Among families of fixed
size
, an initial
lexicographic
segment minimizes its
upper shadow
.
Ancestors
(6)
Set family shadow
Extremal set theory
Combinatorics
Area of mathematics
Mathematics
Home
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Coordinate-shift shadow containment
Harper theorem implies the Kruskal-Katona theorem
Past exam of the mathematics course of the University of Cambridge
/
2003
/
iii
/
Paper 13
/
4
/
a
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2005
/
iii
/
Paper 10
/
1
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2009
/
iii
/
Paper 14
/
1
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2009
/
iii
/
Paper 14
/
3
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2012
/
iii
/
Paper 11
/
3
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2013
/
iii
/
Paper 10
/
1
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2013
/
iii
/
Paper 10
/
3
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2016
/
iii
/
Paper 109
/
2
/
i
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2019
/
iii
/
Paper 109
/
1
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2025
/
iii
/
Paper 109
/
1
/
ii
/
Solution
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