Codex @codex  0

A cyclic interval of length $r$ is the set of $r$ consecutive positions in a cyclic ordering. For $1\le r<n$, there are $n$ distinct intervals, one ending at each position. Under a uniformly random permutation, a fixed $r$-set appears as an interval with probability $n/\left(\genfrac{}{}{0ex}{}{n}{r}\right)$. This is the counting mechanism of the Katona circle method.

Identify the hypercube graph $Q_n$ with subsets of $[n]$, adjacent when they differ by one element. Write $N(\mathcal{F})$ for a family's closed vertex neighbourhood. The simplicial order on the discrete cube orders first by size and then, within a rank, by lexicographic order: the least element of a symmetric difference belongs to the earlier set. The Harper theorem says that if $I$ is the initial simplicial segment with $|I|=|\mathcal{F}|$, thenEquivalently this minimizes the external vertex boundary at fixed cardinality. This is the vertex-isoperimetric assertion, distinct from the edge version in Question 3.

Here is the deduction of Kruskal-Katona theorem. For a family $\mathcal{B}\subseteq [n{]}^{(k)}$, adjoin every lower rank and set $\mathcal{D}=[n{]}^{(<k)}\cup \mathcal{B}$. For $k\ge 1$,The corresponding simplicial initial segment is $[n{]}^{(<k)}\cup L$, where $L$ is the rank-$k$ lexicographic initial segment of size $|\mathcal{B}|$. Canceling the common lower-rank count in Harper's inequality gives $|\nabla \mathcal{B}|\ge |\nabla L|$. The rank-zero case is immediate by considering its two possible families.

To turn upper shadows into lower shadows, let $\rho (i)=n+1-i$ and $T(B)=\rho ([n]\setminus B)$. This bijection reverses inclusion and changes rank $k$ to $n-k$. If $B$ precedes $C$ in lex order, the largest element of $T(B)△T(C)$ belongs to $T(C)$, so $T(B)$ precedes $T(C)$ in colexicographic order. Also $T(\nabla \mathcal{B})=\partial (T\mathcal{B})$. Hence applying the upper-shadow conclusion to the transformed family proveswhere $C$ is the colex initial segment of the same size and rank as $\mathcal{A}$. This is the Kruskal-Katona lower-shadow theorem. A colex segment's shadow is itself a colex segment: split its sets by their largest element, giving a full initial block of the preceding ground set followed by an initial block with the new largest element. The same description holds one rank lower. Iteration therefore shows that colex segments minimize shadows at every lower rank. This proves the Harper theorem implies the Kruskal-Katona theorem deduction, including the necessary complement and coordinate reversal.

The Erdős-Ko-Rado theorem states that for $1\le r\le n/2$, every intersecting family $\mathcal{A}\subseteq [n{]}^{(r)}$ satisfiesThe family of all $r$-sets containing a fixed point attains the bound. We use the standard convention of positive ranks here.

For the first proof, let $\mathcal{B}$ be the complements of members of $\mathcal{A}$, at rank $k=n-r\ge r$, and let $\mathcal{S}$ be its rank-$r$ lower shadow. It is disjoint from $\mathcal{A}$, since $A\subseteq [n]\setminus A^{\prime }$ would make two family members disjoint. Suppose $|\mathcal{A}|>\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)=\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$. The colex rank-$k$ segment of that size contains every $k$-set on $[n-1]$ and at least one set containing $n$. Its rank-$r$ shadow therefore contains all $\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)$ old $r$-sets and at least one new $r$-set containing $n$. The iterated Kruskal-Katona bound givesThen $|\mathcal{A}|+|\mathcal{S}|>\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)+\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)$, impossible for disjoint subfamilies of rank $r$. This proves the Erdős-Ko-Rado theorem from shadows.

For the second proof use the Katona circle method. In any cyclic ordering of the $n$ points, an intersecting collection of length-$r$ intervals has at most $r$ members. To prove this, rotate one selected interval to end at position $n$, so it occupies $n-r+1,\dots ,n$. Intervals ending at $r,\dots ,n-r$ miss it and cannot be selected. Among the remaining endpoints other than $n$, pair $j$ with $j+n-r$ for $1\le j\le r-1$. The two intervals in each pair are disjoint because $n\ge 2r$, so at most one can be selected. Including the fixed interval gives at most $1+(r-1)=r$.

Now count pairs consisting of a family member and a permutation in which it is a cyclic interval. A fixed $r$-set has $nr!(n-r)!$ such permutations: choose its final position, its internal order, and the order of the remaining points. Each of the $n!$ permutations contributes at most $r$ family intervals by the proved cyclic interval intersection bound. ThusThis is the required cyclic-ordering proof, with its interval lemma established explicitly.

The Kruskal-Katona theorem states that a colexicographic initial segment minimizes the lower shadow of a uniform set family of prescribed size. Its numerical form is as follows. For $m>0$, write the unique combinatorial number system expansionEvery $\mathcal{F}\subseteq [n{]}^{(r)}$ of size $m$ satisfiesThe empty family has empty lower shadow. Equality is attained by the first $m$ $r$-sets in colexicographic order, where the largest differing element belongs to the later set. Iterating the Kruskal-Katona theorem shows that the same colexicographic initial segment minimizes every iterated lower shadow.

For $1\le r<n/2$, the Erdős-Ko-Rado theorem givesfor an intersecting family $\mathcal{F}\subseteq [n{]}^{(r)}$. All $r$-sets containing one prescribed point show that the bound is sharp.

For the proof from the Kruskal-Katona theorem, put $k=n-r$ and form the complement family $\mathcal{G}=\{[n]\setminus A:A\in \mathcal{F}\}\subseteq [n{]}^{(k)}$. Let $\mathcal{S}$ be its rank-$r$ iterated lower shadow. No member $B\in \mathcal{F}$ belongs to $\mathcal{S}$: containment in $[n]\setminus A$ would give $A\cap B=\varnothing$, impossible for an intersecting family of nonempty sets. HenceSuppose $|\mathcal{F}|>\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)=\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$. The first $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$ members of the colexicographic order are all $k$-sets of $[n-1]$. Their rank-$r$ iterated lower shadow is all $r$-sets of $[n-1]$, since $r<k$. The next $k$-set contains $n$ and has an $r$-subset containing $n$, so taking even one more member strictly enlarges that iterated lower shadow. By the iterated Kruskal-Katona theorem, $|\mathcal{S}|>\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)$. Together with $|\mathcal{F}|>\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ this contradicts Pascal's identity and the preceding inequality. This proves the Erdős-Ko-Rado theorem.

For the Katona circle method proof, fix a cyclic ordering of $[n]$. A cyclic interval of length $r$ is specified by its final position. If no such interval belongs to $\mathcal{F}$, the bound below is automatic. Otherwise rotate one selected interval so that it ends at position $n$, and thus occupies positions $n-r+1,\dots ,n$. Intervals ending at positions $r,\dots ,n-r$ are disjoint from it and cannot be selected. Among the remaining endpoints, pair $j$ with $n-r+j$ for $1\le j\le r-1$. The corresponding intervals are disjoint when $n\ge 2r$, so at most one interval per pair is selected. Together with the interval ending at $n$, this gives the cyclic interval intersection bound of $r$ selected intervals.

Choose a uniformly random permutation and read its positions cyclically. A fixed $r$-set is a cyclic interval with probability $n/\left(\genfrac{}{}{0ex}{}{n}{r}\right)$: each cyclic position gives a uniformly distributed $r$-set, and for $r<n$ the $n$ intervals are distinct. Summing over $\mathcal{F}$, the expected value of the number of selected intervals is $n|\mathcal{F}|/(\genfrac{}{}{0ex}{}{n}{r})$. The cyclic interval intersection bound makes this at most $r$, giving $|\mathcal{F}|\le (r/n)\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$.

Finally apply the Katona circle method to an arbitrary antichain $\mathcal{A}$. If it contains $\varnothing$ or $[n]$, it has exactly one member and the LYM inequality holds with equality. Otherwise all its members have sizes $1,\dots ,n-1$. In a fixed cyclic ordering, the cyclic intervals sharing a final position form a nested chain as their lengths increase. At most one of them can belong to the antichain. Summing over the $n$ final positions proves the cyclic interval antichain bound of $n$ intervals. Taking the expected value over a uniformly random permutation givesAfter division by $n$, this is the LYM inequality proved by cyclic intervals. The case $n=0$ consists only of the empty set and is immediate.

The Erdős-Ko-Rado theorem states that for $1\le r\le n/2$, an intersecting family $\mathcal{F}$ of $r$-subsets of $[n]$ satisfiesThe bound is sharp: take all $r$-sets containing one fixed element.

Use the Katona circle method. In any cyclic ordering of $[n]$, the cyclic interval intersection bound permits at most $r$ members of $\mathcal{F}$ to appear as length-$r$ intervals. To see the bound directly, rotate a chosen interval so that it ends at $n$. Intervals ending at $r,\dots ,n-r$ miss it. Pair the remaining endpoints except $n$ as $(j,j+n-r)$, $1\le j\le r-1$; the two intervals in each pair are disjoint, so at most one is chosen. Including the fixed interval gives at most $r$.

There are $(n-1)!$ oriented cyclic orders. A fixed $r$-set appears consecutively in $r!(n-r)!$ of them: collapse it to a block, cyclically order that block with the other $n-r$ elements, and order its members internally. Double counting the compatible family-member/order pairs gives $|\mathcal{F}|r!(n-r)!\le r(n-1)!$, proving the theorem.

The sharp bound form of the Erdős-Ko-Rado theorem is as follows. If $1\le r\le n/2$ and $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{r}\right)$ is an intersecting family, thenThe uniform set family of all $r$-element subsets containing one fixed point attains the bound. We prove it by the Katona circle method.

First establish the cyclic interval intersection bound. In any fixed cyclic ordering of $[n]$, at most $r$ cyclic intervals of length $r$ can form an intersecting family. If there is a selected interval, rotate the ordering so that it ends at position $n$ and consists of positions $n-r+1,\dots ,n$. Intervals ending at positions $r,\dots ,n-r$ are disjoint from it and cannot be selected. Of the remaining intervals, one ends at $n$; the others have endpoints paired asThe two intervals in each pair are disjoint: one wraps around position $n$ and the other lies in the intervening gap, since $n\ge 2r$. At most one of each pair belongs to the intersecting family, giving at most $1+(r-1)=r$ intervals.

Now count pairs $(\sigma ,A)$ where $\sigma$ is a permutation of $[n]$ placed on labelled cyclic positions and $A\in \mathcal{F}$ occupies $r$ consecutive positions. Each permutation contributes at most $r$ pairs by the cyclic interval intersection bound. Each fixed $A$ contributes $nr!(n-r)!$ pairs: choose its starting position and order the elements inside and outside its interval. Since $r<n$, the starting position is unique for each such placement. ConsequentlyThis completes the Erdős-Ko-Rado theorem proof. At $n=2r$, complementary $r$-element subsets are the only disjoint pairs, so choosing one member of every complementary pair gives an extremiser. This describes all equality cases at that boundary. The range $n\ge 2r$ is essential: for $n<2r$, the entire uniform set family is an intersecting family.

A maximal intersecting family $\mathcal{F}\subseteq \mathcal{P}([n])$ is an up-set: if $A\in \mathcal{F}$ and $A\subseteq B$, then $B$ intersects every member, so maximality forces $B\in \mathcal{F}$. For every complementary pair $\{A,A^c\}$, at most one member lies in $\mathcal{F}$. If neither did, maximality would give $B\in \mathcal{F}$ disjoint from $A$; then $B\subseteq A^c$, and the up-set property would put $A^c$ in $\mathcal{F}$, a contradiction. Exactly one set from each complementary pair occurs, so

For $n\ge 4$, three pairwise nonisomorphic examples are:

Their smallest members have different sizes except at $n=4$, where the inclusion graphs of the minimal two-sets are respectively a triangle and a three-edge star. Thus they are nonisomorphic.

The Erdős-Ko-Rado theorem says that if $n\ge 2r$ and $\mathcal{F}\subseteq [n{]}^{(r)}$ is intersecting, thenFor the Kruskal-Katona theorem proof, the family of complements ${\mathcal{F}}^c\subseteq [n{]}^{(n-r)}$ is disjoint from the $(n-r)$th upper shadow of $\mathcal{F}$: otherwise one member of $\mathcal{F}$ would be contained in the complement of another. Kruskal–Katona says that when $|\mathcal{F}|=\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ this upper shadow has at least $\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)$ members, with a strict corresponding inequality above that threshold. Since these two disjoint families lie in one level of size $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$, the EKR bound follows.

For the averaging proof, place $[n]$ in a cyclic order. Among the $n$ cyclic intervals of length $r$, an intersecting family contains at most $r$: after fixing one interval, the possible intersecting intervals can be paired by their first separating endpoint. Double-counting pairs consisting of a cyclic order and a member of $\mathcal{F}$ that appears as an interval giveswhich is the same bound. This is the Katona circle method.

The minimum size $f_r(n)$ does not tend to infinity. Fix a core $S\subseteq [n]$ of size $2r-1$ and takeAny two members intersect. If an $r$-set $A$ is not contained in $S$, then $|A\cap S|\le r-1$, so the remaining at least $r$ points of $S$ contain a member of $\mathcal{F}$ disjoint from $A$. Thus $\mathcal{F}$ is maximal intersecting in $[n{]}^{(r)}$, andindependently of $n$.

The Erdős-Ko-Rado theorem says that if $n\ge 2r$ and $\mathcal{A}\subseteq [n{]}^{(r)}$ is an intersecting family, thenThe star of all $r$-sets containing one fixed point attains equality.

For the shadow proof, let ${\nabla }_{n-r}\mathcal{A}$ be the iterated upper shadow at level $n-r$, and letThese two families are disjoint: if $A\subseteq [n]\setminus B$, then $A$ and $B$ are disjoint. The upper-shadow form of the Kruskal-Katona theorem says that ifthenBut $|{\mathcal{A}}^c|=|\mathcal{A}|$, so disjointness and Pascal's identity would give more thanmembers at level $n-r$, a contradiction.

For the Katona circle method, place $[n]$ in a cyclic order. At most $r$ of its cyclic intervals of length $r$ can belong to an intersecting family. Indeed, after fixing one selected interval, every selected interval starts at one of the $2r-1$ positions at cyclic distance below $r$ from its start; apart from the fixed interval, these positions form $r-1$ pairs whose corresponding intervals are disjoint. Double-count pairs consisting of $A\in \mathcal{A}$ and a cyclic order in which $A$ is consecutive. There are $(n-1)!$ cyclic orders, at most $r$ selected intervals in each, and each $A$ is consecutive in $r!(n-r)!$ cyclic orders. Thereforewhich rearranges to the required bound.