Reverse Odd-Town Problem Analysis

• The paper establishes tight bounds for parity-restricted set families, achieving extremal sizes of n for odd n and n-1 for even n.
• It utilizes modular arithmetic, F2 Gram-matrix arguments, and triangular slice-rank techniques to derive sharp intersection bounds.
• The work also examines supersaturation aspects, quantifying the minimum odd intersections forced when classical oddtown limits are exceeded.

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 $\boldsymbol{\alpha}=(0,1)$ (Zou, 2 Jun 2026, Dong et al., 9 Jun 2026). 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 (Wei et al., 2023). Both formulations are reverse counterparts to the classical Oddtown and Eventown theorems, and both now sit inside broader modular and $k$-wise intersection frameworks (Ahmadi et al., 19 Aug 2025).

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 $n$ (Wei et al., 2023). 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$0 (Wei et al., 2023).

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 $2$1
Reverse Odd-Town as Oddtown supersaturation	every set has odd size; $2$2	minimize the number of odd intersections

In the modular town formalism, for integers $2$3 and $2$4, an $2$5-town (mod $2$6) family is a family $2$7 of subsets of $2$8 such that $2$9 for every $(0,1)$0 and $(0,1)$1 for all distinct $(0,1)$2. The quantity

$(0,1)$3

denotes the maximum size of such a family. Reverse Odd-Town is the special case $(0,1)$4, $(0,1)$5, $(0,1)$6 (Zou, 2 Jun 2026).

In the $(0,1)$7-town formalism, for $(0,1)$8, an $(0,1)$9-town is a family $\boldsymbol{\alpha}$0 such that for every $\boldsymbol{\alpha}$1 and every distinct $\boldsymbol{\alpha}$2,

$\boldsymbol{\alpha}$3

The reverse oddtown case is $\boldsymbol{\alpha}$4 (Dong et al., 9 Jun 2026).

For the supersaturation formulation, the key quantity is

$\boldsymbol{\alpha}$5

the number of odd-intersection pairs, equivalently the number of edges in the odd-pair graph $\boldsymbol{\alpha}$6 (Wei et al., 2023).

2. Extremal size for the parity reverse problem

The general off-diagonal theorem for modular towns states that

$\boldsymbol{\alpha}$7

This resolves a conjecture of Veselinov–Marinov, and specializing to $\boldsymbol{\alpha}$8, $\boldsymbol{\alpha}$9, $\boldsymbol{\alpha}=(0,1)$0 gives

$\boldsymbol{\alpha}=(0,1)$1

for the reverse oddtown parameters (Zou, 2 Jun 2026).

The same paper records the exact $\boldsymbol{\alpha}=(0,1)$2 values in a compact table: $\boldsymbol{\alpha}=(0,1)$3 It cites Babai–Frankl for this determination (Zou, 2 Jun 2026). The triangular slice-rank paper gives the same parity-sensitive conclusion from a sharpened modular bound: when $\boldsymbol{\alpha}=(0,1)$4, $\boldsymbol{\alpha}=(0,1)$5, $\boldsymbol{\alpha}=(0,1)$6, and $\boldsymbol{\alpha}=(0,1)$7 is even, one obtains

$\boldsymbol{\alpha}=(0,1)$8

while for odd $\boldsymbol{\alpha}=(0,1)$9 the bound is $n$0 (Ahmadi et al., 19 Aug 2025).

The $n$1-town paper places the same case inside a full asymptotic classification of $n$2. For $n$3, the canonical decomposition gives $n$4 and $n$5, so Theorem 3 yields

$n$6

as $n$7 (Dong et al., 9 Jun 2026).

That paper also states that Johnston–O’Neill determined

$n$8

(Dong et al., 9 Jun 2026). Since the $n$9-town and triangular slice-rank sources explicitly record an odd-$k$0 construction of size $k$1, this suggests that the precise conventions used in the two frameworks should be checked carefully when citing an exact $k$2 value (Zou, 2 Jun 2026, Ahmadi et al., 19 Aug 2025).

3. Extremal constructions and proof mechanisms

A universal lower-bound construction is the fixed-point or “star” family. Fix $k$3 and define

$k$4

Each $k$5 has even size $k$6, and for $k$7,

$k$8

which has odd size $k$9. This gives an $[n]=\{1,2,\dots,n\}$0-town (mod $[n]=\{1,2,\dots,n\}$1) family of size $[n]=\{1,2,\dots,n\}$2, so

$[n]=\{1,2,\dots,n\}$3

for all $[n]=\{1,2,\dots,n\}$4 (Zou, 2 Jun 2026).

When $[n]=\{1,2,\dots,n\}$5 is odd, there is also a co-singleton construction of size $[n]=\{1,2,\dots,n\}$6: $[n]=\{1,2,\dots,n\}$7 Then $[n]=\{1,2,\dots,n\}$8 is even, and for $[n]=\{1,2,\dots,n\}$9,

$\mathcal A\subseteq 2^{[n]}$0

has size $\mathcal A\subseteq 2^{[n]}$1, which is odd. Hence $\mathcal A\subseteq 2^{[n]}$2 is an $\mathcal A\subseteq 2^{[n]}$3-town (mod $\mathcal A\subseteq 2^{[n]}$4) family of size $\mathcal A\subseteq 2^{[n]}$5, proving $\mathcal A\subseteq 2^{[n]}$6 for odd $\mathcal A\subseteq 2^{[n]}$7 in that framework (Zou, 2 Jun 2026).

Two proof strategies recur. The characteristic-zero linear-algebra proof of the off-diagonal bound $\mathcal A\subseteq 2^{[n]}$8 appends a short “square-sum” tail to characteristic vectors in $\mathcal A\subseteq 2^{[n]}$9, 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 (Zou, 2 Jun 2026). In the parity-only setting, the $n$0 Gram-matrix argument considers incidence vectors $n$1 satisfying

$n$2

Since each $n$3 has even weight, $n$4, a codimension-$n$5 subspace, while the Gram matrix is $n$6, so $n$7; this yields $n$8 in the argument recorded there (Dong et al., 9 Jun 2026).

The triangular slice-rank approach rederives Snevily’s modular theorem and sharpens it when $n$9, $2$00, and $2$01. Its key ingredients are a triangular slice-rank lemma for $2$02-tensors, a complementing step that removes one coordinate when $2$03, and polynomial tensors built from inner products of $2$04–$2$05 characteristic vectors (Ahmadi et al., 19 Aug 2025).

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 $2$06. If $2$07 is a family of odd-sized subsets of $2$08 and

$2$09

the supersaturation problem is to lower-bound $2$10 (Wei et al., 2023).

The sharp theorem proved by Wei–Zhao–Zhang–Ge is: $2$11 and this bound is tight for all $2$12 (Wei et al., 2023). This disproves O’Neill’s conjecture that $2$13 in this regime (Wei et al., 2023).

The paper gives explicit tight constructions. At $2$14, the “six-center plus pairs” construction uses the six $2$15-sets

$2$16

together with $2$17 additional pairs

$2$18

The resulting family $2$19 has $2$20 odd-sized sets on $2$21 and satisfies

$2$22

For larger $2$23, adjoining all singletons $2$24 for $2$25 preserves the odd-pair count, yielding $2$26 and $2$27 (Wei et al., 2023).

A second tight family comes from an eventown product construction. If $2$28 and $2$29 is an eventown family, define

$2$30

Then

$2$31

and

$2$32

With $2$33, trimming to size $2$34 again gives families with $2$35 for all $2$36 (Wei et al., 2023).

For larger excess, let $2$37 denote the minimum possible $2$38 over odd-sized $2$39 with $2$40. The paper proves that if $2$41 for some constant $2$42, then

$2$43

It also gives further asymptotic regimes: $2$44 and

$2$45

(Wei et al., 2023).

5. Relation to Eventown and broader reverse supersaturation

The reverse perspective for Eventown mirrors the Oddtown supersaturation problem. If $2$46 consists of even-sized subsets and

$2$47

the problem is again to force a lower bound on $2$48 (Wei et al., 2023).

For sufficiently large $2$49, the strong theorem proved there states that if $2$50, then

$2$51

This matches O’Neill’s explicit construction and extends earlier verification from $2$52 to a wider range (Wei et al., 2023). A universal Fourier-analytic bound valid for all positive integers $2$53 and $2$54 is

$2$55

described there as a “half-strength” bound (Wei et al., 2023).

For very large even-sized families, the same paper studies the density of odd intersections. If $2$56 is fixed and $2$57, let $2$58 be the largest constant such that

$2$59

Then

$2$60

and hence

$2$61

This places reverse oddtown-type questions inside a broader program in which extremal set systems are replaced by dense or supersaturated families (Wei et al., 2023).

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 (Wei et al., 2023).

6. Modular, $2$62-wise, and methodical extensions

The most general modular framework in the data is the $2$63-town (mod $2$64) theory. The off-diagonal theorem

$2$65

contains reverse oddtown as the case $2$66, while the diagonal theorem

$2$67

recovers an Eventown-type benchmark (Zou, 2 Jun 2026). The same paper proves that equality in the diagonal bound is highly restricted and also shows that diagonal behavior can be subtle by establishing $2$68 (Zou, 2 Jun 2026).

A different extension studies generalized reverse odd-town profiles for prime $2$69 and $2$70: $2$71 Using Snevily’s theorem, a complement lemma, and a trace lemma, the triangular slice-rank paper derives bounds on $2$72 by recursively reducing from $2$73-wise constraints to the $2$74 case. In particular, it states

$2$75

and

$2$76

(Ahmadi et al., 19 Aug 2025).

The $2$77-town theory over $2$78 gives a uniform asymptotic classification for all $2$79. Writing $2$80 in the canonical $2$81-basis and defining its level and grade, Theorem 3 states that for $2$82,

$2$83

For reverse oddtown $2$84, this yields $2$85 (Dong et al., 9 Jun 2026). The same paper also studies the symmetric “ville” variant and proves that for the pairwise constraint corresponding to $2$86,

$2$87

(Dong et al., 9 Jun 2026).

Several open directions are explicitly identified. For Oddtown supersaturation, the exact minimum $2$88 is fully determined only for $2$89; for $2$90 the paper gives asymptotic formulas in several regimes but not a complete exact solution (Wei et al., 2023). Structural classification of all extremal families achieving $2$91 remains open (Wei et al., 2023). For Eventown supersaturation, the conjectured lower bound is proved only up to $2$92, while the universal half-strength bound holds for all $2$93 (Wei et al., 2023). In the modular slice-rank setting, Snevily’s conjectured stronger bound $2$94 remains open (Ahmadi et al., 19 Aug 2025). In the $2$95-town and ville frameworks, the wildcard $2$96-versions and the sharp asymptotics of certain unbounded ville cases are also left open (Dong et al., 9 Jun 2026).

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.