Bialostocki 2006 minimum sets forcing monochromatic triangles

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Arie Bialostocki, Mark J. Nielsen, Minimum Sets Forcing Monochromatic Triangles. Ars Combinatoria 81 (2006), 297-303.

For a triangle T, a pair (S,F) with S a finite planar set and F a collection of three-point subsets of S each forming a triangle congruent to T (the paper writes F as a script T) is called T-forcing if every 2-coloring of S yields a monochromatic member of F; combinatorially such a pair is a 3-uniform hypergraph of chromatic number greater than two. The authors define p(T) and m(T) as the minimum |S| and minimum |F| over T-forcing pairs and answer their Questions 3 and 4: the main Theorem shows there is no triangle T with p(T) <= 6, and the minimum value 7 is attained by the triangle T* with angles pi/7, 2pi/7 and 4pi/7 built on the regular 7-gon, whose hypergraph is the known minimal example (the Fano-type 7-point, 7-edge configuration), so min m(T) = 7 as well. Section 3 gives the construction for T* (Example 1) and closes with Example 2, a nine-point subset of the triangular lattice with twelve copies of the right triangle T' with angles pi/6, pi/3, pi/2, which gives p(T') <= 9 and m(T') <= 12. The paper leaves open the Erdos-Graham-Montgomery-Rothschild-Spencer-Straus conjecture that all non-equilateral triangles are 2-Ramsey; equilateral triangles are not, by an alternating-strip coloring. For problem 173 it bears through finite certificates: a T-forcing pair shows that every 2-coloring of the plane contains a monochromatic congruent copy of T, so T is never the exceptional triangle, and it is a small non-2-colorable 3-uniform hypergraph, the kind of finite gadget a SAT/LRAT search would target; the paper gives such pairs only for T* and T' and does not resolve arbitrary triangles.

Source: https://combinatorialpress.com/article/ars/Volume%20081/volume-81-paper-20.pdf. No notice is printed in the scan; the current publisher's copyright policy states "authors retain the copyright to their work. These articles are licensed under an open access Creative Commons CC BY 4.0 license" (https://combinatorialpress.com/copyright-policy/, read 2026-10-02), naming the Creative Commons Attribution 4.0 license with no carve-out for earlier volumes, and the journal page calls the journal Diamond Open Access (https://combinatorialpress.com/ars/, read 2026-10-02); volume 81 was published by the Charles Babbage Research Centre, so whether the policy reaches this 2006 article is unverified.

Bears on. #173

Results to transcribe.

• Theorem: There is no triangle T with p(T) <= 6; every T-forcing pair (S,F) has |S| >= 7.
• Example 1: For the triangle T* with angles pi/7, 2pi/7, 4pi/7 there is a T*-forcing pair with |S| = 7 and |F| = 7, so p(T*) = m(T*) = 7 and the minima in Questions 3 and 4 both equal seven.
• Definition (T-forcing pair): For a finite planar set S and a collection F of three-point subsets of S each congruent to T, the pair (S,F) is T-forcing if every 2-coloring of S makes some member of F monochromatic; equivalently the 3-uniform hypergraph (S,F) has chromatic number greater than two.
• Example 2: For the 30-60-90 right triangle T' with angles pi/6, pi/3, pi/2, a nine-point subset of the triangular lattice with twelve copies of T' is T'-forcing, so p(T') <= 9 and m(T') <= 12.