---
title: Arrow Relation in Extremal Set Theory
url: https://www.emergentmind.com/topics/arrow-relation
type: topic
---

# Arrow Relation in Extremal Set Theory

In extremal set theory, the **arrow relation** is a quantitative statement about how a large family of subsets of \([n]\) must induce a rich system of traces on some smaller ground set. In the standard notation,
\[
(n,m)\rightarrow(a,b),
\]
the meaning is that for every family \(\mathcal F\subseteq 2^{[n]}\) with \(|\mathcal F|\ge m\), there exists a subset \(T\subseteq [n]\) with \(|T|=a\) such that the trace
\[
\mathcal F_{|T}=\{F\cap T:F\in\mathcal F\}
\]
has size at least \(b\). The concept generalizes Sauer–Shelah type shattering statements and connects trace theory, VC-dimension, forbidden configurations, and hypergraph extremal problems [2507.23375].

## 1. Definition and basic interpretation

Let \([n]=\{1,2,\dots,n\}\), and let \(\mathcal F\subseteq 2^{[n]}\). For a set \(T\subseteq [n]\), the **trace** of \(\mathcal F\) on \(T\) is
\[
\mathcal F_{|T}=\{F\cap T:F\in\mathcal F\}.
\]
Thus the trace records all intersection patterns realized by members of \(\mathcal F\) on the coordinates in \(T\).

The arrow notation
\[
(n,m)\rightarrow(a,b)
\]
means that every family \(\mathcal F\subseteq 2^{[n]}\) with \(|\mathcal F|\ge m\) has an \(a\)-element subset \(T\) on which at least \(b\) distinct traces occur. The survey also uses the shorthand \(\mathcal F\rightarrow(a,b)\) when there exists such a set \(T\), and \(\mathcal F\not\rightarrow(a,b)\) when every \(a\)-subset \(T\) satisfies \(|\mathcal F_{|T}|<b\).

If \(|T|=a\), then \(\mathcal F_{|T}\subseteq 2^T\), so at most \(2^a\) patterns can occur. The extreme case \(|\mathcal F_{|T}|=2^a\) is the usual notion of **shattering**: every subset of \(T\) appears as \(F\cap T\) for some \(F\in\mathcal F\). Arrow relations therefore interpolate between weak local richness and full shattering.

## 2. Foundational results and hereditary reduction

The foundational example is the Sauer–Shelah lemma. In arrow notation it states that
\[
\left(n,\,1+\sum_{i=0}^{k-1}\binom ni\right)\rightarrow(k,2^k),
\]
equivalently,
\[
m(n,k,2^k)=\sum_{i=0}^{k-1}\binom ni+1.
\]
This is the threshold forcing some \(k\)-set to be shattered.

A central reduction due to Frankl is that arrow relations may be tested on **hereditary** families. A family \(\mathcal F\) is hereditary if \(F'\subseteq F\in\mathcal F\) implies \(F'\in\mathcal F\). Frankl’s lemma states that the following are equivalent:

1. \((n,m)\rightarrow(a,b)\).
2. Every hereditary family \(\mathcal F\subseteq 2^{[n]}\) with \(|\mathcal F|=m\) satisfies \(\mathcal F\rightarrow(a,b)\).

The reduction is implemented by the **squash** operation \(S_v\), which replaces certain sets \(F\ni v\) by \(F\setminus\{v\}\) when the latter is absent. Repeated squashing transforms an arbitrary family into a hereditary one without increasing traces, because
\[
|S_v(\mathcal F)_{|T}|\le |\mathcal F_{|T}|
\qquad\text{for every }T\subseteq[n].
\]
This is one of the main structural tools in the subject: extremal obstructions to an arrow relation can be taken hereditary.

## 3. Defect Sauer problems and exact thresholds

For fixed \(n,a,b\), the principal threshold parameter is
\[
m(n,a,b)=\min\{m:(n,m)\rightarrow(a,b)\}.
\]
The case \(b=2^a\) is the Sauer–Shelah lemma. The regime \(b<2^a\) is often called the **defect Sauer** problem, and many exact values are known [2507.23375].

| Parameter | Exact value |
|---|---|
| \(m(n,3,7)\) | \(\left\lfloor \frac{n^2}{4}\right\rfloor+n+2\) |
| \(m(n,4,11)\) | \(\operatorname{ex}(n,K_4)+n+2=\left\lfloor \frac{n^2}{3}\right\rfloor+n+2\) |
| \(m(6,4,12)\) | \(24\) |
| \(m(n,4,12)\), \(n\neq 6\) | \(\binom n2+n+2\) |
| \(m(n,4,13)\), \(n\ge 25\) | \(1+\left\lfloor \frac{n+5}{3}\right\rfloor\left\lfloor \frac{n+4}{3}\right\rfloor\left\lfloor \frac{n+3}{3}\right\rfloor\) |

The case
\[
m(n,3,7)=\left\lfloor \frac{n^2}{4}\right\rfloor+n+2
\]
is the prototype of the method. After hereditary reduction, a family \(\mathcal F\not\rightarrow(3,7)\) cannot contain a \(3\)-set, because any \(3\)-set in a hereditary family yields full trace \(8\) on itself. Hence all members have size at most \(2\), and the \(2\)-sets form a triangle-free graph. Mantel’s theorem then bounds their number by \(\lfloor n^2/4\rfloor\). The lower bound is witnessed by
\[
\mathcal F=\binom{[n]}0\cup \binom{[n]}1\cup E(T(n,2)),
\]
where \(T(n,2)\) is the balanced complete bipartite graph.

Several four-vertex cases remain open. For \(m(n,4,9)\), the problem reduces to \(\operatorname{ex}(n,\{C_3^+,C_4\})\), and the survey gives
\[
\left(\frac n2\right)^{3/2}+o(n^{3/2}) \le m(n,4,9)\le \frac12 n^{3/2}+O(n).
\]
For \(m(n,4,14)\) and \(m(n,4,15)\), the values depend on \(\operatorname{ex}_3(n,K_4^{(3)-})\) and \(\operatorname{ex}_3(n,K_4^{(3)})\), with bounds
\[
\frac{2}{7}\cdot \frac{n^3}{6}+o(n^3) \le m(n,4,14)\le 0.286889\cdot \frac{n^3}{6}+o(n^3),
\]
\[
\frac{5}{9}\cdot \frac{n^3}{6}+o(n^3) \le m(n,4,15)\le 0.561666\cdot \frac{n^3}{6}+o(n^3).
\]
These cases show that defect Sauer questions quickly meet difficult hypergraph Turán problems.

## 4. Single-element removal and the asymptotic slope \(m(s)\)

A second major direction studies the relations
\[
(n,m)\rightarrow(n-1,m-s).
\]
Here \(m(n,s)\) denotes the maximum \(m\) for which this implication holds. By hereditary reduction, this is equivalent to the statement that every hereditary family \(\mathcal F\) on \(n\) vertices with \(m\) edges has minimum degree \(\delta(\mathcal F)\le s\).

Two exact finite-\(n\) results are classical:
\[
m(n,0)=n,
\qquad
m(n,1)=\left\lceil \frac32 n\right\rceil.
\]

Frankl proved that for positive integers \(n,d\),
\[
m(n,2^{d-1}-1)\ge \left\lceil \frac{2^d-1}{d}n\right\rceil.
\]
When \(d\mid n\), a block construction gives equality:
\[
m(n,2^{d-1}-1)=\frac{2^d-1}{d}n.
\]

Watanabe and Frankl proved that the limit
\[
m(s)=\lim_{n\to\infty}\frac{m(n,s)}n
\]
exists. They obtained
\[
m(2^{d-1}-2)=\frac{2^d-2}{d},
\qquad
m(2^{d-1})=\frac{2^d-1}{d}+\frac12.
\]

Recent work determines the asymptotic slope near powers of two with high precision. The survey records:

- **Piga and Schülke (2021)**:
  \[
  m(12)=\frac{28}{5},
  \qquad
  m(2^{d-1}-c)=\frac{2^d-c}{d}
  \quad\text{for }1\le c\le d/4.
  \]

- **Li, Ma, Rong (2024)**:
  \[
  m(11)=\frac{53}{10},
  \]
  and for \(d\ge 50\), \(1\le c\le d-1\),
  \[
  m(2^{d-1}-c)=\frac{2^d-c}{d},
  \qquad
  m(2^{d-1}-d)=\frac{2^d-d-\frac12}{d}.
  \]

- **Reiher and Schülke (2025)**:
  \[
  m(2^{d-1}-c)=\frac{2^d-c}{d}
  \qquad (1\le c\le d-1).
  \]

The survey also lists exact values for \(s\le 16\), for example
\[
m(0)=1,\ m(1)=\tfrac32,\ m(2)=2,\ m(3)=\tfrac73,\ m(4)=\tfrac{17}6,\dots,\ m(16)=\tfrac{67}{10}.
\]
The current picture is therefore sharp just below powers of two, but substantially less complete away from that regime.

## 5. Trace functions, VC-dimension, and extremal families

Instead of fixing \(b\) and asking for the minimum \(m\), one may fix \(n,m,a\) and ask for the largest \(b\) guaranteed. For a family \(\mathcal F\subseteq 2^{[n]}\), define
\[
Tr(\mathcal F,a)=\max_{T\in \binom{[n]}a}|\mathcal F_{|T}|,
\]
and
\[
Tr(n,m,a)=\min_{\substack{\mathcal F\subseteq 2^{[n]}\\|\mathcal F|=m}} Tr(\mathcal F,a).
\]
This is exactly the largest \(b\) such that \((n,m)\rightarrow(a,b)\) holds [2507.23375].

A universal lower bound due to Bollobás and Radcliffe is
\[
Tr(n,m,a)\ge m^{a/n}.
\]
They also proved sharper bounds for polynomial-size families and linear-size traces. For integer \(r\ge2\) and \(0<\alpha<1\),
\[
Tr(n,n^r,\alpha n)\ge (1-o(1))n^{\lambda r},
\]
where \(\lambda_0=\log_2(1+\alpha)\), and
\[
\lambda=
\begin{cases}
\frac{\lambda_0}{H(\lambda_0)} & \text{if }\alpha\in(0,\sqrt2-1),\\[4pt]
\lambda_0 & \text{if }\alpha\in[\sqrt2-1,1).
\end{cases}
\]
Kahn, Kalai and Linial proved that for \(0<\alpha,\beta<1\),
\[
Tr(n,\beta 2^n,\alpha n)\ge (1-n^{-c})2^{\alpha n},
\]
where \(c=c(\alpha,\beta)\).

The survey also emphasizes that initial segments of the Boolean lattice do **not** minimize traces in general. Bollobás and Radcliffe constructed, for any \(r\ge2\) and \(0<\varepsilon<1/2\), a hereditary family \(\mathcal H\subseteq 2^{[n]}\) with
\[
|\mathcal H|\ge \sum_{i=0}^r\binom ni
\]
such that
\[
Tr(\mathcal H,n/2)\le \sum_{i=0}^r\binom{n/2}{i} -(1-\varepsilon)2^{-r}\binom nr.
\]
Hence
\[
Tr\left(n,\sum_{i=0}^r\binom ni,\frac n2\right)=o(n^r)
\qquad (r\ge2).
\]

A major recent advance due to Alon, Moshkovitz, and Solomon states that if \(r,\alpha^{-1}\le n^{o(1)}\), then
\[
Tr(n,n^r,\alpha n)=n^{\mu(1-o(1))}
\]
where
\[
\mu=\mu(r,\alpha)=\frac{r+1-\log(1+\alpha)}{2-\log(1+\alpha)}.
\]
Moreover, if \(r=O(1)\) and \(\alpha^{-1}\le (\log n)^{O(1)}\), then
\[
Tr(n,n^r,\alpha n)=\tilde\Theta(n^\mu).
\]
For \(r=2\) and \(\alpha=1/2\),
\[
Tr(n,n^2,n/2)=\tilde\Theta(n^{1.706695\dots}).
\]

On the structural side, traced and strongly traced sets lead to extremal equalities. Let
\[
tr(\mathcal F)=\{T\subseteq[n]: \mathcal F_{|T}=2^T\},
\]
and define the VC-dimension by
\[
VC(\mathcal F)=\max_{T\in tr(\mathcal F)}|T|.
\]
Pajor proved
\[
|\mathcal F|\le |tr(\mathcal F)|.
\]
A set \(T\) is **strongly traced** by \(\mathcal F\) if there exists \(S\subseteq[n]\setminus T\) such that
\[
\{A\cup S:A\in2^T\}\subseteq \mathcal F.
\]
If \(str(\mathcal F)\) denotes the family of strongly traced sets, then Bollobás, Leader, and Radcliffe proved the reverse Sauer inequality
\[
|str(\mathcal F)|\le |\mathcal F|.
\]

A family is **s-extremal** if
\[
|tr(\mathcal F)|=|\mathcal F|.
\]
Bollobás and Radcliffe showed that this is equivalent to several other properties, including
\[
|str(\mathcal F)|=|\mathcal F|
\]
and the uniqueness of the hereditary family obtained from \(\mathcal F\) by repeated squashing. A related strengthening, **order-shattering**, satisfies the exact identity
\[
|\operatorname{osh}(\mathcal F)|=|\mathcal F|.
\]

## 6. Matrix and hypergraph reformulations, and open directions

Arrow relations admit natural reformulations in the language of forbidden \(0\)-\(1\) matrices and hypergraph traces [2507.23375]. Every family \(\mathcal F\subseteq 2^{[n]}\) corresponds to a simple \(0\)-\(1\) incidence matrix; traces then become configurations in submatrices. For a fixed \(0\)-\(1\) matrix \(\mathbf F\), let \(\operatorname{forb}(m,\mathbf F)\) be the maximum number of columns in an \(m\)-rowed simple matrix avoiding \(\mathbf F\) as a configuration.

In this framework, the complete \(k\)-row configuration \(\mathbf K_k\) yields Sauer–Shelah in the form
\[
\operatorname{forb}(m,\mathbf K_k)=\sum_{i=0}^{k-1}\binom mi=\Theta(m^{k-1}).
\]
Related exact results include
\[
\operatorname{forb}(m,\mathbf K_k^s)=\binom m{k-1}+\binom m{k-2}+\cdots+\binom m0,
\]
\[
\operatorname{forb}(m,2\cdot \mathbf K_k)=\binom mk+\binom m{k-1}+\cdots+\binom m0,
\]
and for general \(t\),
\[
\operatorname{forb}(m,t\cdot \mathbf K_k)
=
\operatorname{forb}(m,t\cdot \mathbf K_k^k)
\le
\frac{t-2}{k+1}\binom mk(1-o(1))
+\binom mk+\cdots+\binom m0.
\]

The hypergraph formulation uses induced Berge copies. For a graph \(F\), let \(\operatorname{Tr}_r(F)\) denote the family of \(r\)-uniform induced Berge copies of \(F\). Füredi and Luo proved that for every graph \(F\) with at least one edge,
\[
\operatorname{ex}(n,\operatorname{Tr}_r(F))
=
\Theta\!\left(\max_{2\le s\le r}\{\operatorname{ex}(n,K_s,F)\}\right).
\]
Mubayi and Zhao determined asymptotics for several clique cases: if \(s\in\{3,4\}\) or \(r\in\{s,s+1\}\), then
\[
\operatorname{ex}(n,\operatorname{Tr}_r(K_s))
=
\left(\frac{n}{s-1}\right)^{s-1}+o(n^{s-1}).
\]

The main open problems surveyed in this area are structural as well as enumerative. They include the Frankl–Wang conjecture
\[
\left(n,1+\prod_{0\le i<\ell}\left\lfloor\frac{n+\ell+i}{\ell}\right\rfloor\right)
\rightarrow
(\ell+1,2^{\ell+1}-2^{\ell-1}+1),
\]
Frankl’s conjecture on shattering in antichains, the Mészáros–Rónyai conjecture that every nonempty s-extremal family contains an element whose removal preserves s-extremality, the exact values of \(m(n,4,14)\) and \(m(n,4,15)\), and the determination of \(m(s)\) away from powers of two.

In this form, the arrow relation functions as a unifying language for extremal trace phenomena. It quantifies how global size in \(2^{[n]}\) forces local complexity, and it does so in a way that naturally interfaces with hereditary set systems, VC-theory, Turán-type bounds, forbidden matrices, and induced hypergraph traces.

Source: https://www.emergentmind.com/topics/arrow-relation