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Source. Theorem 3, p. 250, of Paul Erdős and George Purdy, Some extremal problems in geometry, J. Combinatorial Theory 10 (1971), no. 3, 246--252, DOI 10.1016/0097-3165(71)90028-8, as identified on the source card.

Statement

The notation is that of Section 2 (p. 247), recalled on Theorem 1: $G_3^{(2)}(n)$ is the largest number of triangles $X_0X_iX_j$ of one common positive area, for a fixed point $X_0$ and $n$ further distinct points of $E_3$, and $g_3^{(2)}(n)$ is the largest number of triangles of one common positive area spanned by $n$ distinct points of $E_3$.

Theorem 3 (p. 250). For a constant $c$,

$$G_3^{(2)}(n)\le cn^{2-1/3} \qquad\text{and therefore}\qquad g_3^{(2)}(n)\le cn^{3-1/3}.$$

The constant is not made explicit; the proof takes it large enough for the Kővári–Sós–Turán theorem to apply with a fixed $k$.

Proof pointer

Page 250, in outline. Join $X_i,X_j$ when $X_0X_iX_j$ has area $\Delta$. If the graph has more than $cn^{2-1/3}$ edges, the theorem of Kővári, Sós and Turán on Zarankiewicz's problem gives points $Y_1,Y_2,Y_3$ and $Z_1,\ldots,Z_k$ with every $Y_i$ joined to every $Z_j$. Then every $Z_j$ lies at one fixed distance from each of the three lines $X_0Y_i$, that is on three cylinders, which elementary geometry rules out once $k$ exceeds an absolute constant.

The paper adds (p. 250), without proof, that a theorem on generalized graphs gives for example $g_5^{(2)}(n)\le cn^{3-\epsilon}$ for some $0<\epsilon<1$, and $G_k^{(k)}(n)\le c_kn^{k-\epsilon_k}$.

Dependencies

T. Kővári, V. T. Sós and P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50--57 (reference [4] of the paper). Read depth: claims checked; the statement was read on p. 250 and the proof for its structure only.

Bears on

No Erdős problem page cites this result. The planar Problem 1086 is the two-dimensional question; this theorem concerns three dimensions only.