gpt pro 5.4 attempt on Math-720

Erdős Problem #217 (Round 2)

Question recalled. For which $n$ do there exist $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and such that, after some ordering of those distances, the $i$th distance occurs exactly $i$ times?

1. ROUND-2 OBJECTIVE

Path C: exact construction/verification up to the current frontier. Round 1 had only fully verified the cases $n=3$ and $n=4$. The most promising strict advance is therefore to recover and verify the known planar constructions at the current boundary. In this round I derive explicit coordinates for Palásti's $7$-point construction from her geometric description, exactly verify Palásti's $8$-point construction, and extract exact $5$-point and $6$-point crescent configurations from the $8$-point example. The net result is a fully checked existence theorem for every $n=4,5,6,7,8$.

2. Round-1 FOUNDATION USED

I use the following Round-1 facts.

1. Proposition 4.1: the trivial $n=3$ sanity check.
2. Proposition 4.2: an explicit verified $n=4$ configuration.
3. The Round-1 gap statement: no exact planar construction beyond $n=4$ had been verified inside the write-up.

3. NEW INSIGHT / TOOL (ROUND 2)

This round adds three new ingredients.

1. An explicit coordinate derivation for Palásti's $7$-point construction from the original geometric description.
2. An exact symbolic verification of the distance multiplicities and general-position conditions for the $7$-point and $8$-point configurations.
3. A new observation from exact computation: suitable subsets of the $8$-point configuration already realize the cases $n=5$ and $n=6$, whereas no $7$-point subset of that $8$-point configuration is itself a $7$-point crescent configuration.

4. ATTACK PLAN (ROUND 2)

Round 1 had no verified examples past $n=4$. The exact tasks are therefore:

1. derive coordinates for a genuine $7$-point configuration;
2. verify the $8$-point Palásti coordinates exactly;
3. use subsets of the $8$-point configuration to fill the cases $n=5$ and $n=6$;
4. check whether one can obtain $n=7$ simply by deleting one point from the $8$-point construction.

This closes the entire verified small-$n$ range up to the known planar frontier $n=8$.

5. WORK (ROUND 2)

Proposition 5.1 (Palásti's $7$-point crescent configuration in coordinates). Let \[ \begin{array}{rcl} P_1&=&(0,0),\\[1mm] P_2&=&(1,0),\\[1mm] P_3&=&\left(1+\dfrac{\sqrt6}{2},\dfrac{\sqrt2}{2}\right),\\[3mm] P_4&=&\left(\dfrac{1+\sqrt6}{2},\dfrac{\sqrt2+\sqrt3}{2}\right),\\[3mm] P_5&=&\left(\dfrac12,\sqrt2+\dfrac{\sqrt3}{2}\right),\\[3mm] P_6&=&(0,\sqrt2),\\[1mm] Q&=&\left(\dfrac12,\sqrt2-\dfrac{\sqrt3}{2}\right). \end{array} \] Then the set \[ S_7:=\{P_1,P_2,P_3,P_4,P_5,P_6,Q\} \] is a crescent configuration of size $7$ in the plane.

Derivation of the coordinates. Palásti's geometric description starts from a convex hexagon with alternating side lengths \[ 1,\sqrt2,1,\sqrt2,1,\sqrt2 \] and alternating interior angles \[ 150^\circ,90^\circ,150^\circ,90^\circ,150^\circ,90^\circ. \] Fix $P_1=(0,0)$ and $P_2=(1,0)$. Because the exterior turning angles are \[ 30^\circ,90^\circ,30^\circ,90^\circ,30^\circ,90^\circ, \] the six edge directions are \[ 0^\circ,\ 30^\circ,\ 120^\circ,\ 150^\circ,\ 240^\circ,\ 270^\circ. \] Adding the corresponding side vectors gives exactly the six coordinates above. The perpendicular bisector of $P_1P_2$ is $x=\frac12$, and the displayed point $P_5$ lies on it. The condition $P_5Q=\sqrt3$ with $Q$ inside the hexagon forces \[ Q=\left(\frac12,\sqrt2-\frac{\sqrt3}{2}\right). \] Then \[ |P_6Q|=1,\qquad P_6Q\parallel P_4P_3, \] exactly as in Palásti's construction.

Proof that $S_7$ is crescent. A direct exact computation gives the following multiplicity pattern for the six distinct distances: \[ \sqrt{5-\sqrt6}\ \mbox{occurs once}, \] \[ \sqrt{3-\sqrt6}\ \mbox{occurs twice}, \] \[ \sqrt2\ \mbox{occurs three times}, \] \[ 1\ \mbox{occurs four times}, \] \[ \sqrt3\ \mbox{occurs five times}, \] \[ \sqrt{3+\sqrt6}\ \mbox{occurs six times}. \] Thus the distance multiplicities are exactly $1,2,3,4,5,6$.

To verify general position, use the determinant criteria \[ \Delta_{ijk}:=\det\begin{pmatrix} x_i&y_i&1\\ x_j&y_j&1\\ x_k&y_k&1 \end{pmatrix} \] for collinearity, and \[ \Gamma_{ijkl}:=\det\begin{pmatrix} x_i&y_i&x_i^2+y_i^2&1\\ x_j&y_j&x_j^2+y_j^2&1\\ x_k&y_k&x_k^2+y_k^2&1\\ x_l&y_l&x_l^2+y_l^2&1 \end{pmatrix} \] for concyclicity. All these determinants lie in the field $\mathbb{Q}(\sqrt2,\sqrt3)=\mathbb{Q}\oplus \mathbb{Q}\sqrt2\oplus \mathbb{Q}\sqrt3\oplus \mathbb{Q}\sqrt6$. Expanding every determinant in the basis $\{1,\sqrt2,\sqrt3,\sqrt6\}$, an exact symbolic check shows that all \[ \binom73=35 \] collinearity determinants and all \[ \binom74=35 \] concyclicity determinants are nonzero. Therefore no three points are collinear and no four are concyclic. $\square$

Proposition 5.2 (Palásti's $8$-point crescent configuration). Let \[ \begin{array}{rcl} A_1&=&(0,1),\\ A_2&=&(\sqrt3,0),\\ A_3&=&(2\sqrt3,0),\\ A_4&=&\left(\dfrac{5\sqrt3}{2},\dfrac52\right),\\ A_5&=&\left(\dfrac{3\sqrt3}{2},\dfrac92\right),\\ A_6&=&\left(\dfrac{\sqrt3}{2},\dfrac72\right),\\ A_7&=&\left(\dfrac{3\sqrt3}{2},\dfrac72\right),\\ A_8&=&(\sqrt3,2). \end{array} \] Then \[ S_8:=\{A_1,\ldots,A_8\} \] is a crescent configuration of size $8$ in the plane.

Proof. A direct exact computation of the $28$ squared distances gives the multiplicities \[ 1 \ \mbox{once},\qquad 19 \ \mbox{twice},\qquad 21 \ \mbox{three times}, \] \[ 3 \ \mbox{four times},\qquad 4 \ \mbox{five times},\qquad 7 \ \mbox{six times},\qquad 13 \ \mbox{seven times}. \] Thus, after ordering the distinct distances as \[ 1,\ \sqrt{19},\ \sqrt{21},\ \sqrt3,\ 2,\ \sqrt7,\ \sqrt{13}, \] their multiplicities are exactly \[ 1,2,3,4,5,6,7. \]

For general position, the same determinant criteria as in Proposition 5.1 apply. Now all determinants lie in the quadratic field $\mathbb{Q}(\sqrt3)$, so vanishing is equivalent to the vanishing of both coefficients in the basis $\{1,\sqrt3\}$. An exact symbolic check shows that all \[ \binom83=56 \] collinearity determinants and all \[ \binom84=70 \] concyclicity determinants are nonzero. Hence $S_8$ is in general position. $\square$

Proposition 5.3 (exact $5$-point and $6$-point subsets). With the notation of Proposition 5.2, set \[ T_5:=\{A_1,A_2,A_3,A_7,A_8\}, \qquad T_6:=\{A_1,A_2,A_3,A_4,A_7,A_8\}. \] Then $T_5$ is a planar crescent configuration of size $5$, and $T_6$ is a planar crescent configuration of size $6$.

Proof. General position is inherited from $S_8$.

For $T_5$, the exact squared-distance multiplicities are \[ 7 \ \mbox{once},\qquad 3 \ \mbox{twice},\qquad 4 \ \mbox{three times},\qquad 13 \ \mbox{four times}. \] Hence the four distinct distances have multiplicities $1,2,3,4$.

For $T_6$, the exact squared-distance multiplicities are \[ 21 \ \mbox{once},\qquad 3 \ \mbox{twice},\qquad 7 \ \mbox{three times},\qquad 4 \ \mbox{four times},\qquad 13 \ \mbox{five times}. \] Hence the five distinct distances have multiplicities $1,2,3,4,5$.

Therefore $T_5$ and $T_6$ are crescent configurations. $\square$

Corollary 5.4. Planar crescent configurations exist for every \[ n=4,5,6,7,8. \]

Proof. Use the Round-1 verified example for $n=4$, Proposition 5.3 for $n=5,6$, Proposition 5.1 for $n=7$, and Proposition 5.2 for $n=8$. $\square$

Proposition 5.5 (no $7$-point subset of $S_8$ is crescent). No $7$-point subset of the $8$-point configuration $S_8$ from Proposition 5.2 is itself a $7$-point crescent configuration.

Proof. There are only $8$ such subsets. An exact check of the distance multiplicities of each subset shows that none has multiplicity pattern $1,2,3,4,5,6$. Hence a separate $7$-point construction really is needed. $\square$

6. ADVERSARIAL VERIFICATION

The new work was checked against the main possible vulnerabilities.

1. Derivation of the $7$-point coordinates. I verified that the alternating side-length and angle data lead to the displayed edge directions and hence to the displayed vertex coordinates. I also checked directly that $P_5$ lies on the perpendicular bisector of $P_1P_2$, that $P_5Q=\sqrt3$, that $P_6Q=1$, and that $P_6Q\parallel P_4P_3$.
2. Exactness of the checks. For $S_7$, all equalities were checked in the basis $\{1,\sqrt2,\sqrt3,\sqrt6\}$. For $S_8$, all equalities were checked in the basis $\{1,\sqrt3\}$. No floating-point arithmetic was used.
3. Subset inheritance. Because general position is hereditary under taking subsets, only the distance multiplicities needed to be checked for $T_5$ and $T_6$.
4. Need for a separate $7$-point construction. The exact failure of all eight $7$-point subsets of $S_8$ confirms that Proposition 5.1 is not redundant.

7. FINAL

UNRESOLVED (BUT STRICTLY ADVANCED).

The Round-2 advance is substantial: I have now given fully checked planar crescent configurations for every \[ n=4,5,6,7,8. \] This matches the known planar frontier. The remaining open range is therefore \[ n\geq 9. \]

8. COMPLETION ESTIMATE

COMPLETION: 60%

9. REFERENCES

1. I. Palásti, On the seven points problem of P. Erdős, Studia Sci. Math. Hungar. 22 $1987$, 447--448.
2. I. Palásti, Lattice point examples for a question of Erdős, Period. Math. Hungar. 20 $1989$, 231--235.
3. D. Burt, E. Goldstein, S. Manski, S. J. Miller, E. A. Palsson, and H. Suh, Crescent configurations, preprint $2015$, used here for the explicit $8$-point coordinates.
4. T. F. Bloom, Erdős Problem #217, problem page accessed 2026-03-07.