Erdos 1975 extremal problems geometry

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construction_p301: Places 3m points on three circles in orthogonal coordinate planes of six-dimensional space so that they span m^3 congruent equilateral triangles.

theorem_1: Bounds the maximum number of isosceles triangles among n points in the plane between c n^2 log n and c n^{5/2}.

theorem_10: Bounds the number of pairwise congruent triangles among n points in three-dimensional space by cn^{19/9}.

theorem_2: Constructs n points in three-dimensional space with at least 2n^3/27 − cn^2 isosceles triangles.

theorem_3: Bounds the maximum number of equilateral triangles among n points in the plane between n^2/6 − cn^{3/2} and n^2/3.

theorem_4: Bounds the number of equilateral triangles among n points in four-dimensional space by cn^{8/3}.

theorem_5: Bounds the number of pairwise similar triangles among n points in the plane by cn^2.

theorem_6: Bounds the number of pairwise similar triangles among n points in three-dimensional space by cn^{7/3}.

theorem_7: Bounds the number of pairwise similar triangles among n points in four-dimensional space by cn^{17/6}.

theorem_8: Bounds the number of pairwise similar triangles among n points in five-dimensional space by cn^{26/9}.

theorem_9: Shows the number of pairwise congruent triangles among n points in the plane is o(n^{3/2}).

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P. Erdős, G. B. Purdy: Some extremal problems in geometry, III., Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), Congress. Numer. XIV, pp. 291--308, Utilitas Math., Winnipeg, Man., 1975 (MR 52 #13650; Zentralblatt 328.05018).

Erdős and Purdy bound four extremal counts for n points in E^k: isosceles triangles f_k^i, equilateral triangles f_k^e, pairwise congruent triangles f_k^c and pairwise similar triangles f_k^s, all posed at the end of their earlier paper (pp. 291-292). In the plane Theorem 1 (p. 293) gives c_1 n^2 log n < f_2^i(n) < c_2 n^{5/2}, and Theorem 2 (p. 295) gives f_3^i(n) >= 2n^3/27 - cn^2 by a construction in E^3. Theorem 3 bounds planar equilateral triangles by n^2/6 - cn^{3/2} <= f_2^e(n) <= n^2/3 (p. 296), and Theorem 4 gives f_4^e(n) <= cn^{8/3} (p. 299) by the Kovari-Sos-Turan theorem. For similar triangles Theorems 5-8 (pp. 302-305) give f_2^s(n) <= cn^2, f_3^s(n) <= cn^{7/3}, f_4^s(n) <= cn^{17/6} and f_5^s(n) <= cn^{26/9}; for congruent triangles Theorems 9-10 (pp. 305-306) give f_2^c(n) = o(n^{3/2}) and f_3^c(n) <= cn^{19/9}. The methods are incidence counting with points on lines and circles, lattice constructions (the integer lattice for Theorem 1, the triangular lattice for Theorem 3), and Turan-type bounds for graphs and 3-graphs.

In E^6 the paper gives an unnumbered construction (Section 3, pp. 301-302), which it says also appeared in its references [2] and [4]: 3m points X_i = (u_i,v_i,0,0,0,0), Y_i = (0,0,u_i,v_i,0,0), Z_i = (0,0,0,0,u_i,v_i) on three circles in orthogonal coordinate planes span m^3 congruent equilateral triangles, so f_6^e(n), f_6^c(n) and f_6^s(n) are greater than n^3/27 - cn^2. The print takes u_i^2 + v_i^2 = 1 and calls the triangles side one, but points on different unit circles are sqrt(2) apart; radius 1/sqrt(2) gives side one. The concluding section (p. 307) asks whether f_6^e(n) >= n^3/27 - cn^2 is best possible, and whether even f_6^e(n) <= (1/6 - epsilon)n^3 can be shown. It also asks for the limit of f_2^e(n)/n^2 and whether f_2^e(n) <= (1/3 - epsilon)n^2.

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Bears on.

• Problem 755: the problem asks whether n points of R^6 span at most (1/27 + o(1))n^3 unit equilateral triangles. The paper's E^6 construction (pp. 301-302), rescaled to side one, spans at least n^3/27 - cn^2 of them, so the constant 1/27 cannot be lowered; the paper proves no upper bound in E^6 and asks on p. 307 whether its lower bound for equilateral triangles of every size is best possible.

Result pages.

• Theorem 1 (p. 293): isosceles triangles in the plane.
• Theorem 2 (p. 295): isosceles triangles in E^3.
• Theorem 3 (p. 296): equilateral triangles in the plane.
• Theorem 4 (p. 299): equilateral triangles in E^4, with the Remark on pp. 300-301.
• Construction, pp. 301-302: equilateral triangles in E^6 and the question of p. 307.
• Theorem 5 (p. 302): similar triangles in the plane.
• Theorem 6 (p. 302): similar triangles in E^3.
• Theorem 7 (p. 304): similar triangles in E^4.
• Theorem 8 (p. 305): similar triangles in E^5.
• Theorem 9 (p. 305): congruent triangles in the plane.
• Theorem 10 (p. 306): congruent triangles in E^3.

Read status. Claims checked: the statements on the result pages were read clause by clause against the printed pages.

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