The set \[ \{(0,0),(0,1),(0,3),(0,4),(1,6),(2,6),(5,5),(5,6),(6,3),(6,4)\} \] has ten points. Checking its \(\binom{10}{3}=120\) triples shows that the three squared distances in each triple are different. This proves \(I_7\geq10\).

Form a 3-uniform hypergraph on the 49 grid points. An unordered triple is a hyperedge exactly when two of its three squared distances agree. This definition includes equally spaced collinear triples. Direct enumeration gives 2,488 hyperedges, and valid point sets are precisely the independent sets of this hypergraph.

The executable certificate enumerates every increasing eleven-element candidate. A branch is rejected when its new point completes a hyperedge with two selected points. The search reaches no depth-eleven leaf. Since the condition passes to subsets, every valid set with at least eleven points would contain an independent eleven-set. Hence \(I_7\leq10\), and \[ I_7=10. \]

PatternBoost reports the same value \(f(7)=10\) under its explicitly stated possibly-flat convention. Its authors used SAT solvers to prove optimality for small grids. The independent certificate here reproduces the seven-by-seven case.

Section 4.1 of PatternBoost defines \(f(n)\) for subsets of \([n]^2\) containing no possibly-flat isosceles triangle. It states that SAT solvers find optimal constructions for small \(n\), and Figure 11 records \[ f(4)=6,\quad f(5)=7,\quad f(6)=9,\quad f(7)=10. \] This matches the candidate after translating coordinates by one.

The same paper discusses asymptotic bounds and attributes the question independently to Wu, Ellenberg and Jain, and possibly Erdős. A 2019 MathOverflow question states the collinear-inclusive version explicitly.

OEIS A271907 has a similar title and gives 12 at grid size 7. Its illustrated construction at size 8 contains equally spaced collinear triples, so the sequence follows the nondegenerate-triangle convention in practice. It cannot be used for the present candidate. Keeping the convention explicit prevents the two exact values from being merged.

The program orders points first by \(x\), then by \(y\). It enumerates all \(\binom{49}{3}=18{,}424\) triples and marks the 2,488 whose squared-distance multiset has fewer than three values. Serializing each marked triple as its three one-byte vertex indices gives FNV-1a digest `b36e0b83c873e23f`.

The depth-first search adds vertices in increasing order, so each eleven-set has one branch. Every extension is tested against all selected vertex pairs. The only cutoff is the shortage of later vertices needed to reach eleven. The run visits 991,485 admissible nodes and rejects 4,354,134 extensions that complete a forbidden triple. Node counts by selected-set size are \[ [1,39,780,9035,59093,205914,353328,267906,84906,10035,448,0]. \] The final zero is the exhaustive nonexistence result. A running FNV-1a trace of each visited node's depth, next index, and 49-bit selected mask ends at `1ece71b42de90c50`.

The same executable checks all 120 triples in the ten-point witness before searching. Its stable six-line output has SHA-256 digest `ae59f5df2c114c75acb584c00390969aae589e9606cbde7d1d3cd50e0a854519`.