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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0130, E0212, E0213.

Rational and integral realizations

E0130 and E0213: finite blocks and one infinite graph

The following equivalence needs no compactness assumption about coordinate height:

1. There are arbitrarily large finite integral-distance sets with no three collinear and no four cocircular.
2. One countable planar set in that general position has an integer-distance graph with unbounded finite clique sizes.

The reverse implication takes finite cliques and their subsets. For the forward implication, choose integral blocks of sizes tending to infinity and place them one at a time by rigid motions. At each finite stage first rotate the next block so none of its segment directions is parallel to an old segment. Choose its translation outside the following bad sets: point coincidences, mixed collinear triples, mixed cocircular quadruples, and cross-block distances equal to any nonnegative integer.

The first, second and fourth types are points, lines or circles. Every mixed circle condition is a proper polynomial condition in translation. For three old points and one new point, or the reverse, it is a translated circle condition. For two old and two new points, translate the new segment onto a line through one old endpoint, with three distinct points on that line; the other old endpoint is off it by the chosen nonparallel directions. Those four points are not concyclic, proving the polynomial is not identically zero. The mixed collinearity polynomials are likewise proper. The countable union of these closed nowhere-dense algebraic sets cannot fill the plane (or any open region), so a permitted translation exists. This completes the induction.

The resulting integer-distance graph is a disjoint union of finite cliques with unbounded sizes, and contains no infinite complete subgraph. This distinction matters: unbounded clique number is the supremum of finite clique sizes, not the existence of an infinite clique. The held pending infinite-chromatic constructions do not supply these finite integral blocks. The equivalence is a completed author argument; neither side has been established unconditionally. Separately, every planar integer-distance graph has a countable proper coloring: partition the plane into half-open squares of side one half and give each square its own color. Its diameter is less than one, so no two points of the square are at a positive integer distance. This upper bound does not give a finite bound on the chromatic number.

E0212: density implies finite blocks, not a common denominator

If an everywhere dense rational-distance set exists, choose any finite number of points greedily. At each stage avoid the finitely many lines through two earlier points and circles through three earlier noncollinear points. Their complement contains a nonempty open set, so density supplies the next point. The chosen set is in general position. Clear its finitely many distance denominators to get an integral-distance set of the same size. This proves E0212 positive would imply E0213 positive.

The common denominator is finite-set dependent. Applying this procedure to an infinite set and asserting one finite scaling would be invalid; even rational numbers on a line need not have bounded denominators. Solymosi--de Zeeuw Theorem 2.1 permits infinitely many rational-set points on an algebraic curve only if there is a line or circle component. Theorem 2.2 makes all but at most four or three points lie on such a line or circle if its intersection is infinite. A dense set cannot have that form. Decomposing a curve into finitely many components therefore shows that a hypothetical dense rational set meets every real algebraic curve only finitely often. Topological density does not imply an infinite intersection with a closed curve, so there is no contradiction. The primary statements were read; the Faltings-based proof was not rechecked.

E0213: the held bound is a height obstruction

Greenfeld--Iliopoulou--Peluse Corollary 1.3 gives, for an integral general-position set in $[-N,N]^2$, a bound $n\le C(\log N)^A$ with fixed constants. Hence

$$N\ge\exp\bigl((n/C)^{1/A}\bigr).$$

It does not bound $n$ when the diameter is unrestricted. Translation can put a finite configuration inside such a box, but cannot make $N$ uniform in its cardinality. The next discriminating construction remains an eight-point general-position rational or integral set, or a genuine size-only obstruction. The seven-point example is inherited from the exact problem account and was not rerun. The source introduction, Theorem 1.1, Remark 1.2 and Corollary 1.3 were inspected; not the complete number-theoretic proof.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/distance_problems/greenfeld_2024_integer_distance_sets/_index|greenfeld 2024 integer distance sets]]: Introduction; Theorem 1.1; Corollary 1.3 (statement reading). Theorem 1.1; Remark 1.2, bounded-height qualification (statement reading).
• [[../library/distance_problems/solymosi_2010_question_erdos_ulam/_index|solymosi 2010 question erdos ulam]]: Section 2, Theorems 2.1 and 2.2 (statement reading).