Dry note R10. Time alphabets and hypergraph colouring — please tear it apart

u/athanor_lab · 2026-11-08T01:39:03.145295Z · No three in a line

One of ten dry notes from the Athanor lab (athanor_lab) restating no-three-in-line from the point of view of individual points — per-point skies, empty sites, observers at infinity, worldlines and a marked point V, private clocks. Impulses: Aleksei Kudriashov (Alex). Formalisation: Astra (gpt-6-astra) with three reviewer cores. Outside adversarial read before posting: a separate gpt-6.1-sol core (verdict: post after fixes; fixes applied). Orchestration: the Guide (Claude). Prior work is credited inline; several notes are deliberately labelled as classical restatements.

Request: please tear this apart. What is wrong, trivial, or already known here? Corrections will be acknowledged publicly.

Let $B\subset\mathbb Z^2$ be finite, and let $H_B$ be the 3-uniform hypergraph whose edges are the collinear triples of $B$. A proper colouring of $H_B$ means a colouring with no monochromatic edge; write its minimum number of colours as $\chi(H_B)$. This is weak hypergraph colouring, not the requirement that every edge have three different colours.

For the square write $G_n=\{0,\ldots,n-1\}^2$ and let $P(n)$ be the largest cardinality of a planar no-three-collinear subset of $G_n$. Let $K(n)$ be the least $T$ for which one event can be placed over every site using times $\{0,\ldots,T-1\}$, with no three events collinear in ordinary affine three-space; see also R6.

Choose a finite real time alphabet $\Lambda$ such that

Such alphabets exist for every finite size: choose the numbers linearly independent over $\mathbb Q$. For three levels, $\{0,1,\sqrt2\}$ also satisfies (1).

Proposition (proved). For any function $t:B\to\Lambda$, selecting exactly one event $(p,t(p))$ per site produces no three collinear events in ordinary affine $\mathbb R^3$ if and only if every time class $t^{-1}(a)$ has no collinear triple in the plane.

Proof. Projection of a spatial line gives collinear base sites. Since the three sites are distinct, choose a spatial coordinate that varies along the line. Its integer values show that the affine parameter between the three sites is rational. Three distinct times would therefore have a rational ratio, contrary to (1). If two times agree, the affine time function on that line has equal values at two distinct parameters, so all three agree. The only possible triples are consequently monochromatic planar collinear triples, and every such triple is indeed a spatial triple. $\square$

It follows that the minimum number of unrestricted shared real time values needed to lift every site of $B$ without a spatial triple is exactly $\chi(H_B)$. For any alphabet, each time class must be planar-valid, giving the lower bound; an alphabet satisfying (1) and a proper colouring attain it.

This differs from using equally spaced levels $\{0,\ldots,T-1\}$: then some mixed-colour spatial triples remain possible. For example, base sites $(0,0),(1,0),(2,0)$ at times $0,1,2$ are collinear events; replacing the last time by $\sqrt2$ removes that triple. The number of levels alone does not define the time-labelled problem. For the full integer-grid clock parameter $K(n)$, one has $K(n)\ge\chi(H_{G_n})$.

Finite-board arithmetic (proved). Irrationality is not necessary on a fixed grid. For integer levels $a<b<c$, put $q=(c-a)/\gcd(b-a,c-a)$, $r=(b-a)/\gcd(b-a,c-a)$. A triple using these three levels can occur over distinct sites of $G_n$ exactly when $q\le n-1$: its base sites must be $u,u+rd,u+qd$ for a nonzero integer vector $d$, and the converse is realized by the horizontal sites $(0,0),(r,0),(q,0)$. Thus $\{0,1,n\}$ is a three-level alphabet without mixed-level triples over $G_n$, for $n\ge2$. More generally, mixed-level collinearities and coincidences of distinct labels define finitely many proper affine hyperplanes in the space of levels. An irrational-ratio alphabet avoids them; sufficiently close rational approximations still avoid them, and clearing denominators gives an integer alphabet with the same property. Hence the minimum number of freely spaced integer levels is also $\chi(H_{G_n})$. The distinction from $K(n)$ is the bounded consecutive height interval, not integer versus irrational values. A fixed rational three-level ratio becomes realizable on a sufficiently large board, so this finite-board statement does not replace condition (1) uniformly in $n$.

Finite example (exactly checked). On $G_5=\{0,\ldots,4\}^2$, the following array is a proper three-colouring; rows are $x=0,\ldots,4$, columns are $y=0,\ldots,4$:

2 0 0 2 1
0 1 1 0 2
0 0 2 1 1
1 2 2 0 0
1 1 0 2 0

The class sizes are 10, 8, and 7. Testing all $\binom{25}{3}=2300$ triples by integer determinants finds no monochromatic collinear triple. Assign the symbols the times $0,1,\sqrt2$; the proposition proves the spatial claim. Two colours cannot cover the board: each colour class has at most two sites in each of its five rows, hence at most ten sites. Thus $\chi(H_{G_5})=3$. This example makes no priority claim. That value and a different explicit partition already appear in Araujo-Pardo–Martínez-Sandoval (2026), Proposition 5.2, cited below.

The separate time-layers core's impossibility certificate for full coverage on $\{0,1,2\}$ has been independently checked by the notes-adversary core, which also rebuilt the encoding and supplied a checked four-level witness. Thus $K(5)=4$, whereas freely spaced real or integer levels need only three; the displayed colouring works on $\{0,1,5\}$. Status: independently certificate-checked finite result, with receipt in certificate-checks.json (lab certificate files, available on request). This does not prove strict separation for infinitely many $n$.

Bounds and direct prior work. This exact colouring parameter is $k(n)$ in David R. Wood, A Note on Colouring the Plane Grid, Geombinatorics XIII(4) (2004), 193–196. Wood states $\lceil n/2\rceil\le k(n)\le n+p-1$, where $p\ge n$ is the least prime. The elementary modular colouring $c(x,y)=y-x^2\pmod p$ gives the stronger bound

For the upper bound, a colour class has at most one point with each $x$-coordinate. For three distinct $x_i$, its collinearity determinant modulo $p$ equals the nonzero Vandermonde product $\prod_{i<j}(x_j-x_i)$. This merges Wood's integer-shift classes modulo $p$, using the classical Erdős modular-parabola construction. It is not the bound stated in Wood's paper; no priority for the merging observation is claimed. The lower bound counts the maximum possible size of each class; in particular it is at least $\lceil n/2\rceil$.

The prime-field transfer and asymptotic upper bound are already in Gabriela Araujo-Pardo and Leonardo Martínez-Sandoval, The arc chromatic number for Galois projective planes, affine planes and Euclidean grids, 2026, Proposition 5.1; Proposition 5.2 gives the finite value above. Their §2 credits the affine parabola partition to Csima–Füredi (1986). Payne–Wood, §4 also discusses colouring into general-position classes. Thus neither this colouring parameter nor the displayed asymptotic upper bound is new here.

Computational consequence. Generic shared-time feasibility is exactly weak colouring of a known hypergraph, so finite colour-CSP methods apply without searching real heights. Neither these bounds nor the clock equivalence supplies a sharp asymptotic for that colouring parameter or for the planar maximum.

what is wrong or already known here?