[proofplan] We combine three ingredients: the [Existence of Fully Compressed Family](/theorems/2595) theorem, which produces a family $\mathcal{B}$ of the same size as $\mathcal{A}$ with shadow no larger and which is fully compressed; the [Characterisation of Colex Initial Segments](/theorems/2594) theorem, which identifies fully compressed families as exactly the colex initial segments; and the uniqueness of the colex initial segment of a given cardinality. Together, these show that $\mathcal{B}$ is the unique colex initial segment $\mathcal{C}$ of size $|\mathcal{A}|$, and the shadow inequality follows. [/proofplan] [step:Apply the existence theorem to obtain a fully compressed family with controlled shadow] Let $\mathcal{A} \subseteq X^{(r)}$ be an arbitrary family. By the [Existence of Fully Compressed Family](/theorems/2595) theorem, there exists $\mathcal{B} \subseteq X^{(r)}$ satisfying: - $|\mathcal{B}| = |\mathcal{A}|$, - $|\partial \mathcal{B}| \leq |\partial \mathcal{A}|$, - $\mathcal{B}$ is $(U,V)$-compressed for all disjoint $U, V \subseteq X$ with $|U| = |V|$ and $\max V > \max U$. [guided] The goal of the Kruskal--Katona theorem is to show that the colex initial segment of size $|\mathcal{A}|$ achieves the minimum shadow. The strategy is indirect: rather than computing the shadow of $\mathcal{A}$ directly, we replace $\mathcal{A}$ with a better-structured family $\mathcal{B}$ that is easier to analyse. The [Existence of Fully Compressed Family](/theorems/2595) theorem is a pure existence result. Starting from any family $\mathcal{A} \subseteq X^{(r)}$, it produces a family $\mathcal{B}$ that is a fixed point of all valid general compressions $C_{UV}$ with $|U| = |V|$, $U \cap V = \varnothing$, and $\max V > \max U$. The construction works by choosing, among all families of the same size as $\mathcal{A}$ with shadow no larger, one that minimises the potential function $\Sigma(\mathcal{B}) = \sum_{B \in \mathcal{B}} \sum_{i \in B} 2^i$. Any nontrivial compression $C_{UV}$ with $\max V > \max U$ would strictly decrease $\Sigma$ (since it replaces higher-weighted elements from $V$ with lower-weighted elements from $U$), so the minimiser must be a fixed point of all such compressions. The three properties listed above are the only facts about $\mathcal{B}$ that we use in the remainder of the proof. The specific identity of $\mathcal{B}$ -- which sets it contains -- is irrelevant; only the preservation of size, the non-increase of the shadow, and the full compression property matter. [/guided] [/step] [step:Identify $\mathcal{B}$ as the colex initial segment $\mathcal{C}$ via the characterisation theorem] By the [Characterisation of Colex Initial Segments](/theorems/2594) theorem, a family $\mathcal{F} \subseteq X^{(r)}$ is an initial segment of $X^{(r)}$ in the colex order if and only if $\mathcal{F}$ is $(U,V)$-compressed for every pair of disjoint sets $U, V \subseteq X$ with $|U| = |V|$ and $\max V > \max U$. Since $\mathcal{B}$ satisfies this compression condition, $\mathcal{B}$ is a colex initial segment. The colex order on $X^{(r)}$ is a total order, so for each positive integer $m$ there is exactly one initial segment of cardinality $m$. Let $\mathcal{C}$ denote the unique colex initial segment with $|\mathcal{C}| = |\mathcal{A}|$. Since $|\mathcal{B}| = |\mathcal{A}| = |\mathcal{C}|$ and both $\mathcal{B}$ and $\mathcal{C}$ are colex initial segments of the same size, we conclude $\mathcal{B} = \mathcal{C}$. [guided] Why does full compression characterise colex initial segments? The [Characterisation of Colex Initial Segments](/theorems/2594) theorem establishes that the two properties are equivalent: - **Forward direction**: if $\mathcal{F}$ is a colex initial segment and $A \in \mathcal{F}$ satisfies $A \cap (U \cup V) = V$, then $C_{UV}(A) = (A \setminus V) \cup U$. Since $\max V > \max U$, the largest element in $A \triangle C_{UV}(A) = U \triangle V$ belongs to $V \subseteq A$, so $C_{UV}(A) <_{\mathrm{colex}} A$. Because $\mathcal{F}$ is an initial segment, $C_{UV}(A) \in \mathcal{F}$, confirming that $\mathcal{F}$ is $(U,V)$-compressed. - **Converse**: if $\mathcal{F}$ is $(U,V)$-compressed for all valid pairs but is not a colex initial segment, there exist $B \in \mathcal{F}$ and $C \notin \mathcal{F}$ with $C <_{\mathrm{colex}} B$. Setting $U = C \setminus B$ and $V = B \setminus C$ yields a valid compression pair (with $\max V > \max U$ since $C <_{\mathrm{colex}} B$), and $C_{UV}(B) = C \notin \mathcal{F}$, contradicting the assumption that $\mathcal{F}$ is $(U,V)$-compressed. Since $\mathcal{B}$ is fully compressed, the characterisation applies directly and identifies $\mathcal{B}$ as a colex initial segment. The uniqueness of $\mathcal{C}$ follows from the fact that the colex order is a total order on $X^{(r)}$. A total order has exactly one initial segment of each cardinality $m$ (namely the first $m$ elements in the ordering). Since $\mathcal{B}$ and $\mathcal{C}$ are both colex initial segments of cardinality $|\mathcal{A}|$, they must be identical: $\mathcal{B} = \mathcal{C}$. [/guided] [/step] [step:Conclude the shadow inequality $|\partial \mathcal{A}| \geq |\partial \mathcal{C}|$] From the existence theorem, $|\partial \mathcal{B}| \leq |\partial \mathcal{A}|$. From the identification $\mathcal{B} = \mathcal{C}$, we substitute to obtain \begin{align*} |\partial \mathcal{C}| = |\partial \mathcal{B}| \leq |\partial \mathcal{A}|. \end{align*} Since $\mathcal{A} \subseteq X^{(r)}$ was an arbitrary family and $\mathcal{C}$ is the unique colex initial segment with $|\mathcal{C}| = |\mathcal{A}|$, this shows that among all families of a given size, the colex initial segment achieves the smallest possible lower shadow. This completes the proof of the Kruskal--Katona theorem. [guided] Let us trace the chain of inequalities explicitly. We have three facts: 1. $|\partial \mathcal{B}| \leq |\partial \mathcal{A}|$ (from the [Existence of Fully Compressed Family](/theorems/2595) theorem), 2. $\mathcal{B} = \mathcal{C}$ (from the [Characterisation of Colex Initial Segments](/theorems/2594) theorem and the uniqueness of colex initial segments of a given cardinality), 3. $|\mathcal{C}| = |\mathcal{A}|$ (by construction of $\mathcal{C}$). Combining (1) and (2): \begin{align*} |\partial \mathcal{A}| \geq |\partial \mathcal{B}| = |\partial \mathcal{C}|. \end{align*} This is the desired conclusion. The argument is entirely structural: we never computed any shadow sizes explicitly. Instead, the compression machinery did the work -- it guaranteed the existence of a shadow-minimising family (the fully compressed one) and the characterisation theorem identified that family as the colex initial segment. Note that the proof also establishes uniqueness of the extremal family: the colex initial segment $\mathcal{C}$ is the unique minimiser, because the colex order is a total order and thus determines exactly one initial segment of each cardinality. [/guided] [/step]