Post-ductal coarctation places a tight waist in the aorta just beyond the left subclavian artery, so the body must build a detour to carry blood from the high-pressure arch territory down to the low-pressure descending aorta. The detour relies on vessels that branch off the subclavian arteries above the block and anastomose with vessels that empty into the aorta below the block.
The workhorse of this detour is the link between the internal thoracic artery and its anterior intercostal branches on one side and the posterior intercostal arteries on the other. The scapular anastomosis adds extra capacity, with the suprascapular and subscapular arteries enlarging and feeding blood toward the intercostal system. The enlarged, tortuous posterior intercostal arteries are what erode the lower borders of the ribs and produce the classic rib notching on a chest film. So the suprascapular, subscapular and posterior intercostal arteries are all real collateral routes.
The vertebral artery does the opposite job: it heads upward to supply the hindbrain and has no role in shunting blood around the aortic narrowing into the descending aorta. It is the odd one out.
$\therefore$ the vessel that does not form a collateral in post-ductal coarctation is the vertebral artery.
\[\boxed{\text{Vertebral artery}}\]