---
title: Reverse Odd-Town Problem Analysis
url: https://www.emergentmind.com/topics/reverse-odd-town-problem
type: topic
---

# Reverse Odd-Town Problem Analysis

Reverse Odd-Town Problem denotes two closely related set-intersection questions that arise as reversals of the classical Oddtown paradigm. In one usage, it is the parity extremal problem for families of even-sized sets with every pairwise intersection odd; this is the \((a,b)\)-town (mod \(2\)) case \((0,1)\), and also the \(\boldsymbol{\alpha}\)-town case \(\boldsymbol{\alpha}=(0,1)\) [2606.03613, 2606.11139]. In another usage, it is the supersaturation problem for Oddtown: once a family of odd-sized subsets exceeds the classical Oddtown maximum \(n\), how many pairs with odd intersections are forced [2302.05586]. Both formulations are reverse counterparts to the classical Oddtown and Eventown theorems, and both now sit inside broader modular and \(k\)-wise intersection frameworks [2508.13809].

## 1. Definitions and competing formulations

Classically, an oddtown family on \([n]=\{1,2,\dots,n\}\) is a family \(\mathcal A\subseteq 2^{[n]}\) in which every set has odd cardinality and all pairwise intersections are even; Berlekamp’s bound states that the maximum size of such a family is at most \(n\) [2302.05586]. An eventown family is defined analogously with every set of even cardinality and all pairwise intersections even, and Graver’s bound gives maximum size at most \(2^{\lfloor n/2\rfloor}\) [2302.05586].

Two reverse viewpoints are used in the literature.

| Formulation | Family constraint | Extremal question |
|---|---|---|
| Reverse Odd-Town as a parity town | every set has even size; every pairwise intersection has odd size | maximize \(|\mathcal F|\) |
| Reverse Odd-Town as Oddtown supersaturation | every set has odd size; \(|\mathcal A|\ge n+s\) | minimize the number of odd intersections |

In the modular town formalism, for integers \(n\ge k\ge2\) and \(0\le a,b\le k-1\), an \((a,b)\)-town (mod \(k\)) family is a family \(F\) of subsets of \([n]\) such that \(|A|\equiv a\pmod{k}\) for every \(A\in F\) and \(|A\cap B|\equiv b\pmod{k}\) for all distinct \(A,B\in F\). The quantity
\[
m_{k,n}(a,b)
\]
denotes the maximum size of such a family. Reverse Odd-Town is the special case \(k=2\), \(a=0\), \(b=1\) [2606.03613].

In the \(\boldsymbol{\alpha}\)-town formalism, for \(\boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_k)\in\mathbb F_2^k\), an \(\boldsymbol{\alpha}\)-town is a family \(\mathcal F\subseteq 2^{[n]}\) such that for every \(i\in\{1,\dots,k\}\) and every distinct \(F_1,\dots,F_i\in\mathcal F\),
\[
|F_1\cap\cdots\cap F_i|\equiv \alpha_i \pmod 2.
\]
The reverse oddtown case is \(\boldsymbol{\alpha}=(0,1)\) [2606.11139].

For the supersaturation formulation, the key quantity is
\[
op(\mathcal A):=\big|\{\{A,B\}\subset\mathcal A:A\ne B,\ |A\cap B|\ \text{is odd}\}\big|,
\]
the number of odd-intersection pairs, equivalently the number of edges in the odd-pair graph \(H(\mathcal A)\) [2302.05586].

## 2. Extremal size for the parity reverse problem

The general off-diagonal theorem for modular towns states that
\[
a\not\equiv b\pmod{k}\qquad\Longrightarrow\qquad m_{k,n}(a,b)\le n.
\]
This resolves a conjecture of Veselinov–Marinov, and specializing to \(k=2\), \(a=0\), \(b=1\) gives
\[
m_{2,n}(0,1)\le n
\]
for the reverse oddtown parameters [2606.03613].

The same paper records the exact \(k=2\) values in a compact table:
\[
m_{2,n}(0,1)=
\begin{cases}
n, & \text{if } n \text{ is odd},\\
n-1, & \text{if } n \text{ is even}.
\end{cases}
\]
It cites Babai–Frankl for this determination [2606.03613]. The triangular slice-rank paper gives the same parity-sensitive conclusion from a sharpened modular bound: when \(p=2\), \(L=\{1\}\), \(K=\{0\}\), and \(n\) is even, one obtains
\[
|\mathcal F|\le \binom{n-2}{0}+\binom{n-2}{1}=n-1,
\]
while for odd \(n\) the bound is \(n\) [2508.13809].

The \(\boldsymbol{\alpha}\)-town paper places the same case inside a full asymptotic classification of \(f_{\boldsymbol{\alpha}}(n)\). For \(\boldsymbol{\alpha}=(0,1)\), the canonical decomposition gives \(\operatorname{lv}(\boldsymbol{\alpha})=1\) and \(\operatorname{grd}(\boldsymbol{\alpha})=1\), so Theorem 3 yields
\[
f_{(0,1)}(n)=(1+o(1))\,n
\]
as \(n\to\infty\) [2606.11139].

That paper also states that Johnston–O’Neill determined
\[
f_{(0,1)}(n)=n-1\qquad\text{for all }n\ge1
\]
[2606.11139]. Since the \((a,b)\)-town and triangular slice-rank sources explicitly record an odd-\(n\) construction of size \(n\), this suggests that the precise conventions used in the two frameworks should be checked carefully when citing an exact \(k=2\) value [2606.03613, 2508.13809].

## 3. Extremal constructions and proof mechanisms

A universal lower-bound construction is the fixed-point or “star” family. Fix \(x\in[n]\) and define
\[
S_i=\{x,i\}\qquad (i\in[n]\setminus\{x\}).
\]
Each \(S_i\) has even size \(2\), and for \(i\ne j\),
\[
S_i\cap S_j=\{x\},
\]
which has odd size \(1\). This gives an \((0,1)\)-town (mod \(2\)) family of size \(n-1\), so
\[
m_{2,n}(0,1)\ge n-1
\]
for all \(n\) [2606.03613].

When \(n\) is odd, there is also a co-singleton construction of size \(n\):
\[
F_i=[n]\setminus\{i\}\qquad (i\in[n]).
\]
Then \(|F_i|=n-1\) is even, and for \(i\ne j\),
\[
F_i\cap F_j=[n]\setminus\{i,j\}
\]
has size \(n-2\), which is odd. Hence \(\{F_i:i\in[n]\}\) is an \((0,1)\)-town (mod \(2\)) family of size \(n\), proving \(m_{2,n}(0,1)=n\) for odd \(n\) in that framework [2606.03613].

Two proof strategies recur. The characteristic-zero linear-algebra proof of the off-diagonal bound \(m_{k,n}(a,b)\le n\) appends a short “square-sum” tail to characteristic vectors in \(\mathbb Q^{n+N}\), using Lagrange’s four-square theorem, and shows that any nontrivial integer relation among the extended vectors contradicts a gcd condition; this avoids reduction modulo primes and applies uniformly to composite moduli [2606.03613]. In the parity-only setting, the \(\mathbb F_2\) Gram-matrix argument considers incidence vectors \(v_1,\dots,v_m\in\mathbb F_2^n\) satisfying
\[
v_i\cdot v_i\equiv0,\qquad v_i\cdot v_j\equiv1\ \ (i\ne j).
\]
Since each \(v_i\) has even weight, \(v_i\in U=\{x\in\mathbb F_2^n:x\cdot 1\equiv0\}\), a codimension-\(1\) subspace, while the Gram matrix is \(G=J+I\), so \(\operatorname{rank}(G)\ge m-1\); this yields \(m\le n-1\) in the argument recorded there [2606.11139].

The triangular slice-rank approach rederives Snevily’s modular theorem and sharpens it when \(K=\{0\}\), \(L=-L\), and \(p\mid n\). Its key ingredients are a triangular slice-rank lemma for \(2\)-tensors, a complementing step that removes one coordinate when \(p\mid n\), and polynomial tensors built from inner products of \(0\)–\(1\) characteristic vectors [2508.13809].

## 4. Reverse Odd-Town as Oddtown supersaturation

A second, distinct formulation asks for the minimum number of odd intersections forced once an oddtown family exceeds the classical extremal threshold \(n\). If \(\mathcal A\) is a family of odd-sized subsets of \([n]\) and
\[
|\mathcal A|\ge n+s,
\]
the supersaturation problem is to lower-bound \(op(\mathcal A)\) [2302.05586].

The sharp theorem proved by Wei–Zhao–Zhang–Ge is:
\[
\text{if }1\le s\le n-4\text{ and }|\mathcal A|\ge n+s,\ \text{ then }\ op(\mathcal A)\ge s+2,
\]
and this bound is tight for all \(s\le n-4\) [2302.05586]. This disproves O’Neill’s conjecture that \(op(\mathcal A)\ge 3s\) in this regime [2302.05586].

The paper gives explicit tight constructions. At \(n=s+4\), the “six-center plus pairs” construction uses the six \(3\)-sets
\[
C=\{\{1,2,3\},\{1,2,4\},\{1,2,5\},\{1,3,4\},\{1,3,5\},\{1,4,5\}\},
\]
together with \(s-1\) additional pairs
\[
\{i+5\},\ \{2,3,4,5,i+5\}\qquad (i=1,\dots,s-1).
\]
The resulting family \(\mathcal A_s\) has \(2s+4\) odd-sized sets on \([s+4]\) and satisfies
\[
op(\mathcal A_s)=s+2.
\]
For larger \(n\ge s+4\), adjoining all singletons \(\{i\}\) for \(i\in\{s+5,\dots,n\}\) preserves the odd-pair count, yielding \(|\mathcal A|=n+s\) and \(op(\mathcal A)=s+2\) [2302.05586].

A second tight family comes from an eventown product construction. If \([n]=[m]\sqcup\{m+1,\dots,n\}\) and \(\mathcal E\subseteq 2^{[m]}\) is an eventown family, define
\[
\mathcal A(\mathcal E,m,n)=\{E\cup\{i\}:E\in\mathcal E,\ i\in\{m+1,\dots,n\}\}.
\]
Then
\[
|\mathcal A(\mathcal E,m,n)|=|\mathcal E|(n-m)
\]
and
\[
op\big(\mathcal A(\mathcal E,m,n)\big)=\binom{|\mathcal E|}{2}(n-m).
\]
With \(\mathcal E=\{\emptyset,\{1,2\}\}\), trimming to size \(n+s\) again gives families with \(op=s+2\) for all \(1\le s\le n-4\) [2302.05586].

For larger excess, let \(M_o(s,n)\) denote the minimum possible \(op(\mathcal A)\) over odd-sized \(\mathcal A\) with \(|\mathcal A|=n+s\). The paper proves that if \(s=cn+o(n)\) for some constant \(c>0\), then
\[
M_o(s,n)=\Big(\binom{\lfloor c\rfloor+1}{2}+(\lfloor c\rfloor+1)(c-\lfloor c\rfloor)\Big)n+o(n).
\]
It also gives further asymptotic regimes:
\[
M_o(s,n)=\tfrac12\,\frac{s^2}{n}+\tfrac12\,s+o(s)\quad\text{if }s=o(n^2/\log n)\text{ and }n=o(s),
\]
and
\[
M_o(s,n)=\tfrac12\,\frac{s^2}{n}+o\!\left(\frac{s^2}{n}\right)\quad\text{if }\log s=o(n)\text{ and }n=o(s)
\]
[2302.05586].

## 5. Relation to Eventown and broader reverse supersaturation

The reverse perspective for Eventown mirrors the Oddtown supersaturation problem. If \(\mathcal A\subseteq 2^{[n]}\) consists of even-sized subsets and
\[
|\mathcal A|\ge 2^{\lfloor n/2\rfloor}+s,
\]
the problem is again to force a lower bound on \(op(\mathcal A)\) [2302.05586].

For sufficiently large \(n\), the strong theorem proved there states that if \(s\le 2^{\lfloor n/8\rfloor}/n\), then
\[
op(\mathcal A)\ge s\cdot 2^{\lfloor n/2\rfloor-1}.
\]
This matches O’Neill’s explicit construction and extends earlier verification from \(s=1,2\) to a wider range [2302.05586]. A universal Fourier-analytic bound valid for all positive integers \(n\) and \(s\) is
\[
op(\mathcal A)> s\cdot 2^{\lfloor n/2\rfloor-2},
\]
described there as a “half-strength” bound [2302.05586].

For very large even-sized families, the same paper studies the density of odd intersections. If \(\epsilon\in(0,1/2)\) is fixed and \(|\mathcal A|\ge 2^{(1-\epsilon)n}\), let \(f_n(\epsilon)\) be the largest constant such that
\[
op(\mathcal A)\ge f_n(\epsilon)\binom{|\mathcal A|}{2}.
\]
Then
\[
\tfrac12\Big(1-2^{(\epsilon-1/2)n}\Big)\le f_n(\epsilon)\le
\begin{cases}
\tfrac12, & n\text{ odd},\\[4pt]
\tfrac12\Big(1-\frac{1}{2^{n-1}-1}\Big), & n\text{ even},
\end{cases}
\]
and hence
\[
\lim_{n\to\infty} f_n(\epsilon)=\tfrac12.
\]
This places reverse oddtown-type questions inside a broader program in which extremal set systems are replaced by dense or supersaturated families [2302.05586].

The comparison is conceptually important because classical Oddtown/Eventown results say how large a family can be without any odd intersections, whereas the reverse supersaturation questions quantify unavoidable odd intersections once those extremal bounds are exceeded [2302.05586].

## 6. Modular, \(k\)-wise, and methodical extensions

The most general modular framework in the data is the \((a,b)\)-town (mod \(k\)) theory. The off-diagonal theorem
\[
a\not\equiv b\pmod{k}\Longrightarrow m_{k,n}(a,b)\le n
\]
contains reverse oddtown as the case \(m_{2,n}(0,1)\le n\), while the diagonal theorem
\[
m_{k,n}(a,a)\le 2^{\lfloor n/2\rfloor}
\]
recovers an Eventown-type benchmark [2606.03613]. The same paper proves that equality in the diagonal bound is highly restricted and also shows that diagonal behavior can be subtle by establishing \(m_{3,11}(2,2)>m_{3,11}(1,1)\) [2606.03613].

A different extension studies generalized reverse odd-town profiles for prime \(p\) and \(k\ge3\):
\[
\alpha=(0,\ldots,0,L),\qquad L\subseteq\mathbb Z_p,\quad L=-L,\quad 0\notin L.
\]
Using Snevily’s theorem, a complement lemma, and a trace lemma, the triangular slice-rank paper derives bounds on \(|\mathcal F|\) by recursively reducing from \(k\)-wise constraints to the \(k=2\) case. In particular, it states
\[
|\mathcal F|\le \sum_{i=0}^{|L|}\binom{\left\lfloor\frac{n}{2^{\,k-2}}\right\rfloor - 2}{i} + (k-2)\qquad (p\mid n),
\]
and
\[
|\mathcal F|\le \sum_{i=0}^{|L|}\binom{\left\lfloor\frac{n}{2^{\,k-3}}\right\rfloor - 2}{i} + (k-2)\qquad (p\nmid n)
\]
[2508.13809].

The \(\boldsymbol{\alpha}\)-town theory over \(\mathbb F_2\) gives a uniform asymptotic classification for all \(k\). Writing \(\boldsymbol{\alpha}\) in the canonical \(\gamma\)-basis and defining its level and grade, Theorem 3 states that for \(\operatorname{lv}(\boldsymbol{\alpha})=t>0\),
\[
f_{\boldsymbol{\alpha}}(n)=
\begin{cases}
(1+o(1))\,(t!)^{1/t}\,n^{1/t}, & \operatorname{grd}(\boldsymbol{\alpha})=1,\\[4pt]
(1+o(1))\,(t!/2)^{1/t}\,n^{1/t}, & \operatorname{grd}(\boldsymbol{\alpha})=2.
\end{cases}
\]
For reverse oddtown \(\boldsymbol{\alpha}=(0,1)\), this yields \(f_{(0,1)}(n)=(1+o(1))n\) [2606.11139]. The same paper also studies the symmetric “ville” variant and proves that for the pairwise constraint corresponding to \(\alpha=(0,1)\),
\[
g_{(0,1)}(n)\le2\qquad\text{for all }n
\]
[2606.11139].

Several open directions are explicitly identified. For Oddtown supersaturation, the exact minimum \(op(\mathcal A)\) is fully determined only for \(1\le s\le n-4\); for \(s>n-4\) the paper gives asymptotic formulas in several regimes but not a complete exact solution [2302.05586]. Structural classification of all extremal families achieving \(op(\mathcal A)=s+2\) remains open [2302.05586]. For Eventown supersaturation, the conjectured lower bound is proved only up to \(s\le 2^{\lfloor n/8\rfloor}/n\), while the universal half-strength bound holds for all \(n,s\) [2302.05586]. In the modular slice-rank setting, Snevily’s conjectured stronger bound \(|\mathcal F|\le\binom{n}{s}\) remains open [2508.13809]. In the \(\boldsymbol{\alpha}\)-town and ville frameworks, the wildcard \(*\)-versions and the sharp asymptotics of certain unbounded ville cases are also left open [2606.11139].

These developments show that Reverse Odd-Town is no longer a single isolated extremal problem. It is now a nexus joining parity-restricted set systems, modular intersection theory, supersaturation, slice-rank arguments, and higher-order intersection frameworks.

Source: https://www.emergentmind.com/topics/reverse-odd-town-problem