Suppose $n\in\mathbb{N}.$ Partition $S_n:=\{1,2,3,\ldots,2n\}$ into $n$ pairs. Is it true that the maximum pair product is $\geq n^2 ?\ $ Or $\geq n(n+1)?$
So for example for $n=3$, we have: $S_3:=\{1,2,3,4,5,6\}.$ If we partition it like this: $(1,6), (2,5),\ (3,4),\ $ the largest pair product is $3\times 4 = 12 > 9.$ We can easily check by brute force that however we partition $S_3$ into three pairs, the maximum of all pair products will be $>3^2 =9.$ In fact it will be $\geq 3\times 4 = 12.$
If we try to look for a counter-example to the conjecture, we write pairs starting with the largest number in the first entry, then the second number is the greatest number such that that pair product is $<n^2$ and continue down, then for example, for $n=15,$ ($15^2 = 225)$, we get: $(30, 7),\ (29, 6),\ (28, 8),\ (27, 5),\ (26, 4),\ (25, 3),\ (24, 9),\ (23, 2),\ (22, 10),\ (21, 1),\ (20, 11),\ $ but now we have to do $(19,12)$ and $19\times 12 > 225.$
I suspect the propositions are probably true, but I don't see how to actually prove it.