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Statement

Notation (p. 11). $\mathbb E^N\xrightarrow{2}(X_1,X_2)$ means that for every partition $\mathbb E^N=C_1\cup C_2$, either $C_1$ contains a congruent copy of $X_1$ or $C_2$ contains a congruent copy of $X_2$; a crossed arrow denies it. $P_2$ is a set of two points at distance $1$.

The chapter lists, unnumbered, as a sampling of results of this type (p. 11, items (i) to (v)):

(i) $\mathbb E^2\xrightarrow{2}(T_2,T_3)$, where $T_i$ is any subset of $\mathbb E^2$ with $i$ points, $i=2,3$.

(ii) $\mathbb E^2\xrightarrow{2}(P_2,P_4)$, where $P_4$ is a set of four collinear points with distance $1$ between consecutive points.

(iii) $\mathbb E^3\xrightarrow{2}(T,Q^2)$, where $T$ is an isosceles right triangle and $Q^2$ is a square.

(iv) $\mathbb E^2\xrightarrow{2}(P_2,T_4)$, where $T_4$ is any set of four points [Juh79]. So for every partition of the plane into $C_1$ and $C_2$, either $C_1$ contains two points at distance $1$ or $C_2$ contains a congruent copy of $T_4$.

(v) There is a set $T_8$ of $8$ points such that $\mathbb E^2$ does not arrow $(P_2,T_8)$ [CT94]: some partition of the plane into $C_1$ and $C_2$ has no two points of $C_1$ at distance $1$ and no congruent copy of $T_8$ in $C_2$. The chapter says this strengthens an earlier result of Juhász [Juh79], who proved it for a certain set of $12$ points.

The chapter introduces items (i) to (iii) without individual citations and points to [EGM+73], [EGM+75a] and [EGM+75b] for more results of this type.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of the Handbook of Discrete and Computational Geometry, 2nd edition, CRC Press (2004), read in the preprint of the chapter identified on the source card, whose own page number is cited: the notation and items (i) to (v) on p. 11, in the subsection "Asymmetric Ramsey theorems" of Section 11.6. Item (iv) cites R. Juhász, Ramsey type theorems in the plane, J. Combin. Theory Ser. A 27 (1979), 152--160; item (v) cites G. Csizmadia and G. Tóth, Note on a Ramsey-type problem in geometry, J. Combin. Theory Ser. A 65 (1994), 302--306.

Read depth. Claims checked: the notation and each item were read clause by clause on the page image of the preprint. The chapter gives no proofs, and the cited papers were not read for this page. Nothing here is independently reviewed.

Proof pointer

No proof is printed; the items are cited to the papers named above.

Dependencies

None in the chapter.

Bears on

• Problem 214: item (iv) reports Juhász's theorem that in every partition of the plane into $C_1$ and $C_2$ with no two points of $C_1$ at distance $1$, $C_2$ contains a congruent copy of every four-point set, in particular of the four vertices of a unit square, which is the problem's question. Item (v) reports the eight-point set of Csizmadia and Tóth and Juhász's earlier twelve-point set, for which the corresponding statement fails.
• Problem 188: item (ii) states that every partition of the plane into $C_1$ and $C_2$ with no two points of $C_1$ at distance $1$ has four collinear points of $C_2$ with consecutive distances $1$, that is a four-term progression with a step of length $1$. The chapter gives no colouring that avoids longer such progressions and states no value of the problem's $k$.