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Statement

Setting (p. 124). $X_3$ is the set of three collinear points with consecutive distances $1$, and $\lfloor\cdot\rfloor$ is the floor function.

Example (p. 124, unnumbered). Partition $\mathbb{E}^N$ into four classes $C_i=\{\bar x:\lfloor\bar x\cdot\bar x\rfloor\equiv i \pmod 4\}$. Then for every integer $t$, no class $C_i$ contains a congruent copy of $(2t+1)X_3$.

The print indexes the classes by $1\le i\le 4$ in the definition and by $i=0,1,2,3$ in the proof; the four residues modulo 4 are meant either way.

The paper presents the example as a strengthening of Bourgain's set of positive upper density containing no congruent copy of $t_iX_3$ along a sequence $t_1<t_2<\cdots$ tending to infinity (pp. 122, 124). It adds (p. 125) that the same argument applies to a nonspherical set whose coefficients $c_i'$ in the linear dependence of the proof of the theorem on p. 122 are all rational, and that the analogous statement for every nonspherical set was not then known.

Proof pointer

Pp. 124--125. For a copy $\{x,y,z\}$ of $(2t+1)X_3$ with $y$ the middle point, the law of cosines gives $x\cdot x+z\cdot z-2\,y\cdot y=2(2t+1)^2$. Writing each squared norm as $4M+i+\varepsilon$ with $0\le\varepsilon<1$ and using $(2t+1)^2\equiv1\pmod 8$ leads to $4M+\varepsilon_x-2\varepsilon_y+\varepsilon_z=2$ for an integer $M$, which the bounds on the $\varepsilon$'s rule out.

Read depth

Claims checked: the statement, its page and the indexing of the classes were read clause by clause on the print. The proof was read for structure only. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. R. L. Graham, Recent trends in Euclidean Ramsey theory, Discrete Math. 136 (1994), 119--127, doi:10.1016/0012-365X(94)00110-5; the edition read is named on the source card.

Bears on

No Erdős problem directly. The partition concerns odd integer dilates of three collinear points in a space of any dimension; the paper does not relate it to a listed problem.