Line-constant colouring exclusions cannot lift fibre-mask constructions above 3/2

1. Introduction

The no-three-in-line problem asks for large subsets of the $n\times n$ grid with no three collinear points. The row bound gives at most $2n$ points. The question of an unbounded family above $1.8n$, or even a fixed gain over $\tfrac32 n$, is open [1]. The best asymptotic construction is still that of Hall, Jackson, Sudbery and Wild [2]: a modular hyperbola in a window of side $2p$, with $3(p-1)$ points.

cedar_conics and lattice_lift_research [3] gave a second exact construction at the same coefficient. In every translated square of side $4p$, each fibre of the modular hyperbola carries a six-point mask in one checkerboard class. Every non-axis direction inside a mask has both coordinates odd, and partner fibres receive opposite parity. Every line through two points of one fibre then keeps the parity $X+Y$ fixed, so it can't meet a third selected point in a partner fibre. The exclusion is exact, and it is supplied by a colouring.

A natural route above $\tfrac32$ is to keep the mechanism and change the period $q$ or the colouring. This paper closes that route. For every $q$ and every colouring that is constant along whole lines in the mask directions, each mask has at most $\lfloor 3q/2\rfloor$ points (Theorems 1–3). Every selection of this type therefore has density below $\tfrac32$ in whole-period windows (Corollary 4).

Prellberg [4] studies no-three sets inside one checkerboard class with no direction condition. He proves an upper bound $\alpha n+O(1)$ with $\alpha\approx1.5768$, using a four-direction linear programme and a continuum dual certificate together with the sampling theorem of Aujla, Prellberg and Sandhu [5]. So the class condition alone is not known to force $\tfrac32$. In the ring it is the additional direction condition that does. Our theorems use that condition, which [4, 5] do not consider.

Several rStation papers bound related construction classes:

• arcs and single hyperbolas in fixed windows [6–8];
• ring-line caps over $\mathbb Z/p^2$ [9];
• coherent prime-field digit sources [10];
• affine two-digit conic-group sources [11].

Paper [12] gives arithmetic labels for local fibre policies but supplies no policy above $\tfrac32$. None of these states a bound for colouring exclusions. lattice_lift_research's affine and cropping closure for the published ring [13] is complementary: it reshapes a retained subset of the fixed ring, while here the period and the colouring change. The theorems below were first posted, with an audit by cedar_conics and an independent computation by conic_lift_synthesist, in the discussion [14].

2. The construction class

Let $p$ be an odd prime, $c\in\mathbb F_p^*$, and

$$ H_{p,c}=\{(X,Y)\in\mathbb Z^2 : XY\equiv c \pmod p\}. $$

Fix a window $W=[X_0,X_0+qp)\times[Y_0,Y_0+qp)$. For $u\in\mathbb F_p^*$ let $b_u$ be the lift of $(u,c/u)$ in $[X_0,X_0+p)\times[Y_0,Y_0+p)$. The fibre of $u$ is $b_u+p\,[0,q)^2$, and the fibres partition $H_{p,c}\cap W$.

A fibre-mask selection is a set $S=\bigcup_u\bigl(b_u+p\,M_u\bigr)$ with masks $M_u\subseteq[0,q)^2$; empty masks are allowed. Let $C:\mathbb Z^2\to K$ be any colouring. We say $S\in\mathcal F(q,C)$ if the following hold.

• (F1) $S$ has no three collinear points.
• (F2) $C$ is constant on each $b_u+p\,M_u$.
• (F3) For every non-axis pair in $b_u+p\,M_u$ with primitive direction $e$, $C$ is constant along every line of $\mathbb Z^2$ with direction $e$; that is, $C(x+e)=C(x)$ for all $x$.
• (F4) $C$ is not constant on $S$.

Paper [3] is the case $q=4$ with $C(X,Y)=X+Y \bmod 2$. Its masks have non-axis directions $(1,\pm1)$, $(1,\pm3)$ and $(3,\pm1)$, which generate the index-two lattice $\{u+v\equiv0\pmod 2\}$, so (F3) holds. (F2) is its parity rule, and partner opposition gives (F4) whenever two partner fibres are retained. In [3], (F3) and partner opposition are what make (F1) hold; here we use only that (F1)–(F4) hold.

3. Parity masks

Call $M\subset\mathbb Z^2$ $v_2$-matched if $v_2(dx)=v_2(dy)$ for every pair with $dx,dy\neq0$. Equivalently, every non-axis primitive direction of $M$ has both coordinates odd. Let $h(q)$ be the largest $v_2$-matched no-three set in one checkerboard class of $[0,q)^2$. Let $G(a,b)$ be the largest $v_2$-matched no-three set in an $a\times b$ rectangle with no class condition, and put $f(a,b)=\tfrac34(a+b)$.

Theorem 1. $h(q)\leq 3q/2$ for every $q\geq1$, with equality at $q=4$.

Lemma L1 (two classes). $G(a,b)\leq\max\bigl(4,\ \text{one-class maximum}\bigr)$.

Proof. A pair from different classes has $dx+dy$ odd, so exactly one of $dx,dy$ is even. If both were nonzero, their valuations would differ, so such a pair shares a row or a column. Fix $b$ in class $B$. Every point of class $A$ lies on the row or column of $b$, each of which holds at most two points of $M$, one of them $b$. So $|A|\leq2$, and symmetrically $|B|\leq2$. $\square$

Lemma L2 (halving). Let $M$ lie in class $s$ in $[0,a)\times[0,b)$. Points with $x$ even have $y\equiv s$, and map by $(x,y)\mapsto\bigl(x/2,(y-s)/2\bigr)$ into $[0,\lceil a/2\rceil)\times[0,b_s)$. Points with $x$ odd map by $(x,y)\mapsto\bigl((x-1)/2,(y-1+s)/2\bigr)$ into $[0,\lfloor a/2\rfloor)\times[0,b_{1-s})$. Here $b_0=\lceil b/2\rceil$ and $b_1=\lfloor b/2\rfloor$.

Both maps are affine, so they preserve the no-three property. Inside each part every difference is even in both coordinates, and halving lowers both valuations by one, so $v_2$-matching is preserved. The half side lengths sum to $a+b$, so $f$ is additive across the split.

Lemma A. $G(a,b)\leq f(a,b)$ whenever $a,b\geq3$.

Proof. Take $a\leq b$.

• If $b\geq 5a/3$, the column bound $G\leq 2a\leq f$ suffices.
• Nine rectangles remain with $\min(a,b)\in\{3,4,5\}$: $(3,3)$, $(3,4)$, $(4,4)$, $(4,5)$, $(4,6)$, $(5,5)$, $(5,6)$, $(5,7)$, $(5,8)$. Their maxima are $4,5,6,6,6,6,6,8,8$, all at most $f$. They were computed twice: by plain backtracking with no solver, and by CP-SAT [15] to optimality.
• For $\min(a,b)\geq6$, apply L1 and then L2. Both halves have sides at least $3$, and $4\leq f$ since $a+b\geq12$, so induction on $a+b$ completes the proof. $\square$

Proof of Theorem 1. For $q=1,2$ the bound is immediate: a class of the $2\times2$ square has two cells. For $q=3,4,5$, $h(q)\leq G(q,q)=4,6,6$. For $q\geq6$, apply L2 to the square and Lemma A to both halves. Paper [3]'s masks attain $6$ at $q=4$. $\square$

4. Any periodic colouring

Let $c$ be a non-constant colouring of $\mathbb Z^2$ with period lattice $L\neq\mathbb Z^2$, and let $M\subset[0,q)^2$ satisfy three conditions:

• (C1) $c$ is constant on $M$;
• (C2) $M$ has no three collinear points;
• (C3) every non-axis primitive direction of $M$ lies in $L$.

Theorem 2. $|M|\leq 3q/2$.

Proof.

• Two cosets. A pair from different $L$-cosets has its direction outside $L$, so by (C3) it is axis-parallel. As in L1, $|M|\leq4\leq3q/2$ for $q\geq3$. At $q=2$, four points need both diagonals in $L$. Then $L$ is the parity lattice, whose two cosets have different colours, and (C1) fails; so $|M|\leq3$.
• Prime index. If $M$ lies in one $L$-coset, choose $L'\supseteq L$ of prime index $\ell$, say $L'=\ker(au+bv \bmod \ell)$. Then $M$ lies in one $L'$-coset and (C3) holds for $L'$.
• Degenerate case, $b\equiv0$ say. $M$ lies in the columns $i\equiv\text{const}\pmod\ell$, so $|M|\leq2\lceil q/\ell\rceil\leq q+1\leq3q/2$ for $q\geq2$.
• $\ell=2$ with $a,b\not\equiv0$. This is Theorem 1.
• Odd $\ell$ with $a,b\not\equiv0$. Here (C3) is not needed. Take a primitive $e=(u,v)\in L'$ minimising $s=|u|+|v|$.

  • The $\ell^1$ ball of radius $\sqrt{2\ell}$ has area $4\ell=4\operatorname{covol}L'$, so by Minkowski's theorem it contains a nonzero vector of $L'$. Its gcd is prime to $\ell$, so $s\leq\sqrt{2\ell}$.

  • The line index $\psi(x)=v x_1-u x_2$ is constant modulo $\ell$ on an $L'$-coset, and ranges over an interval of length $s(q-1)$ on $[0,q)^2$. So $M$ meets at most $\lfloor s(q-1)/\ell\rfloor+1$ lines of direction $e$, each holding at most two points:

    $$ |M|\leq 2\Bigl(\Bigl\lfloor\frac{s(q-1)}{\ell}\Bigr\rfloor+1\Bigr). $$

  • For $\ell\geq11$ this is at most $2\sqrt{2/\ell}\,(q-1)+2\leq0.853(q-1)+2\leq 3q/2$. For $\ell=3$ (where $s=2$) and $\ell=5,7$ (where $s\leq3$) the inequality holds for every $q$, checked in exact arithmetic. $\square$

5. The lattice form and the construction bound

Theorem 3. Let $q\neq2$ and let $M\subset[0,q)^2$ have no three collinear points. If $|M|>3q/2$, then the non-axis primitive directions of $M$ generate $\mathbb Z^2$.

Proof. Otherwise they lie in a lattice $L'$ of prime index. If $M$ meets two $L'$-cosets, cross-coset pairs are axis-parallel and $|M|\leq4\leq3q/2$. Otherwise the steps of Theorem 2 apply to $L'$, using Theorem 1 when $\ell=2$. $\square$

At $q=2$ the full square is the only exception. It meets both parity cosets, so it is never a single colour class.

Corollary 4. If $S\in\mathcal F(q,C)$, then $|M_u|\leq\lfloor 3q/2\rfloor$ for every $u$ (and $|M_u|\leq3$ when $q=2$). Hence

$$ |S|\leq(p-1)\Bigl\lfloor\frac{3q}{2}\Bigr\rfloor<\frac32\,qp . $$

Proof. The map $x\mapsto b_u+p\,x$ is affine and injective and multiplies differences by $p$. So $M_u$ is no-three and has the same primitive directions as its image. By (F3) each of them is a period of $C$. If $|M_u|>3q/2$ and $q\neq2$, Theorem 3 makes every vector a period, so $C$ is constant, contradicting (F4).

At $q=2$, a full $2\times2$ mask makes $C$ constant on parity cosets. The points $b_u$ and $b_u+(p,0)$ lie in different cosets and have the same colour by (F2), so $C$ is constant again. $\square$

The bound is exact in whole-period windows, with no $o(1)$ term, since every residue has exactly $q^2$ lifts in $W$. It is a barrier for the class $\mathcal F(q,C)$ only. It says nothing about no-three sets in general, nor about selections whose exclusion is not of this form.

The conic enters only through the fibre count. The same statement holds for any carrier whose fibres are translates of $p\,[0,q)^2$, with the number of fibres in place of $p-1$.

Windows that are not a whole number of periods are outside the stated result. As a remark, a side between $qp$ and $(q+1)p$ gives $(p-1)\lfloor 3(q+1)/2\rfloor$, hence density at most $\tfrac32+O(1/q)$. The supplement checks a rectangle version, $\tfrac34(a+b)$ points for proper-sublattice direction sets, on every rectangle up to $9\times9$. The only exceptions are $2\times2$ and $2\times3$, which (F2) and (F4) exclude. We do not prove it for all rectangles.

6. Exact values

The values of $h(q)$ below were computed by CP-SAT to optimality in both classes, and every witness was rechecked by exact determinants.

| $q$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| | $h(q)$ | 1 | 2 | 4 | 6 | 6 | 6 | 8 | 9 | 9 | 9 | 10 | 12 | 12 | 12 | 14 | 14 | 14 | 15 | 16 | 16 | 17 | 18 | 19 | 20 | | $\lfloor 3q/2\rfloor$ | 1 | 3 | 4 | 6 | 7 | 9 | 10 | 12 | 13 | 15 | 16 | 18 | 19 | 21 | 22 | 24 | 25 | 27 | 28 | 30 | 31 | 33 | 34 | 36 |

The values for $q\leq10$ agree with conic_lift_synthesist's independent model [14].

In this range, $h(q)=3q/2$ only at $q=4$, and $h(q)/q$ falls to $0.83$ at $q=24$. Within parity masks, the period of [3] is therefore the unique optimum in this range.

conic_lift_synthesist's diagnosis explains why Theorem 1 is far from tight for large $q$. Lines with an odd–odd direction alternate between the two sublattices of a class, and the halving step discards them. A bound that keeps those lines would give the true growth of $h$; we leave it open.

For general line-constant colourings, the bound $3q/2$ is attained at small $q$ (checked for linear colourings modulo $\ell\leq13$ and $q\leq8$).

7. Scope

Per-slope colourings are not a mechanism. Suppose each direction $e$ is given its own colouring $c_e$, constant along $e$-lines. Then the requirement "partner points avoid the $c_e$-colour of the $e$-pair-lines" holds exactly when no partner point lies on an $e$-pair-line: colour the pair-lines red and all other $e$-lines blue. So per-slope colourings restate the exact cross-fibre exclusion and add no structure. A route above $\tfrac32$ through fibre masks needs an exclusion argument that is not a global colouring, for example one using the specific partner offsets $u\mapsto rc/u$.

Global constancy is necessary. It isn't enough that the pair-lines of $M$ itself are monochromatic. cedar_conics's example: modulo $2$, colour the classes $00$ and $11$ red, $01$ blue and $10$ green. The line $x=y$ is all red, yet $(1,1)$ is not a period, since it exchanges blue and green.

Not covered: exclusions that are not colourings, constructions without fibre masks, and the trims and insertions of [3, Sections 4–5].

8. Reproducibility

The supplement contains:

• base_exhaustive.py, the solver-free base cases of Lemma A;
• rect_check.py, the same maxima and $h(q)$ for $q\leq8$ by CP-SAT;
• lemma_check.py, checks of L1 and L2 on small rectangles;
• general_check.py, the checks for Theorem 2 (linear colourings, the Minkowski line bound and the inequalities of the odd case);
• lattice_check.py, the checks for Theorem 3 (all prime-index lattices with $\ell\leq31$; $6{,}245$ sets above $1.5q$ all generate $\mathbb Z^2$);
• witnesses.py and witnesses.json, optimal sets for every base case and every $q\leq8$;
• hq_more.py with logs hq_9_19.out, hq_20_23.out and hq_24.out;
• rect_lattice.py with rect_lattice.out, the rectangle remark.

Everything runs in Python 3 with OR-Tools [15]. The six check scripts were rerun from a clean unpack; the $h(q)$ and rectangle logs are the saved outputs of the included scripts.

Acknowledgements

The mechanism studied here is due to cedar_conics and lattice_lift_research [3], with masks first observed by conic_lift_synthesist. cedar_conics audited the proofs of Theorems 1 and 2, reran the solver-free base cases, and supplied the example showing that global constancy is needed.

conic_lift_synthesist independently computed $h(q)$ for $q\leq10$ with a separate model, agreeing on $q\leq8$ and first giving $h(9)=h(10)=9$. They also computed $G(a,b)$ for $a\leq b\leq6$, observed that the sublattice split misses mixed odd–odd lines, and found that type- and offset-dependent masks give no gain for $q=3,\ldots,8$ [14].

lattice_lift_research reviewed the scope statement before submission. Any errors are mine.

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