Every positive integer $m$ has a unique <binomial representation>
$$
m=\binom{a_r}{r}+\binom{a_{r-1}}{r-1}+\cdots+\binom{a_s}{s},
\qquad a_r>a_{r-1}>\cdots>a_s\geq s\geq1.
$$
Define
$$
f_r(m)=\binom{a_r}{r-1}+\binom{a_{r-1}}{r-2}+\cdots+\binom{a_s}{s-1}.
$$
The <Kruskal-Katona theorem> says that every $r$-uniform family $\mathcal A$ of size $m$ has
$$
|\partial\mathcal A|\geq f_r(m),
$$
and the initial segment of length $m$ in <colexicographic order> attains equality.

We prove the inequality by simultaneous <mathematical induction> on the ground-set size and on $r$. We use the elementary <binomial-shadow arithmetic lemma>
$$
f_r(a+b)\leq\max\{f_r(a),b\}+f_{r-1}(b).
$$
To verify the lemma, greedily remove the largest term $\binom qr$ from the binomial representation of $a+b$. If $a$ reaches that term first, apply the induction hypothesis to the remainders; otherwise transfer the excess to $b$. In the crossing case <Pascal's identity>
$$
\binom qr=\binom{q-1}r+\binom{q-1}{r-1}
$$
gives exactly the two terms on the right. This induction also proves uniqueness of the greedy representation.

Choose the largest ground-set element $x$ and split
$$
\mathcal A_0=\{A\in\mathcal A:x\notin A\},
\qquad
\mathcal A_1=\{A\setminus\{x\}:x\in A\}.
$$
The members of the lower shadow that avoid $x$ form $\partial\mathcal A_0\cup\mathcal A_1$, while those containing $x$ are the sets $B\cup\{x\}$ with $B\in\partial\mathcal A_1$. Hence, writing $a=|\mathcal A_0|$ and $b=|\mathcal A_1|$,
$$
|\partial\mathcal A|
=|\partial\mathcal A_0\cup\mathcal A_1|+|\partial\mathcal A_1|
\geq\max\{f_r(a),b\}+f_{r-1}(b)
\geq f_r(a+b).
$$
This completes the induction.

Finally, the initial colex segment first contains all $r$-subsets of $[a_r]$, followed recursively by sets containing $a_r+1$ whose remaining elements form the appropriate initial segment at level $r-1$. Its shadow has the same recursive decomposition, and therefore has exactly $f_r(m)$ members. This proves sharpness.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 109 / 1 / i / Solution

Let $f_r(m)$ be the lower-shadow size prescribed by the binomial representation in the <Kruskal-Katona theorem>. Then
$$
f_r(a+b)\leq\max\{f_r(a),b\}+f_{r-1}(b).
$$
Induction using <Pascal's identity> proves the inequality and drives the standard ground-set induction for the theorem.

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 Incoming links: Binomial-shadow arithmetic lemma