my biggest 3.5 lbs
1. Use a raised Bed in full sun lots of Humus.
2. Plant extra deep.
3. Stake and prune all suckers
4. Use liquid fertilizer.
5. Mulch in summer heat.
6. Water a little more often.
7 let only first 2 fruit on each cluster set.
8 When fruit reaches 1"-2 inches diameter thin cluster to just one fruit.
9. After 4 clusters set don't let any more clusters set.
10. When fruit are 3"-4" thin plant to 2 or 3 fruit total.

My biggest single fruit was just over 2 1/2 Lb. I use raised beds but only pinch back suckers while the vine is low to the ground to ease my trellising effort. I do use raised beds and lots of organic material with a permanent mulch of leaves or pine needles or what ever I can rustle up. I don't thin fruit.

I don't know that I have even thought about using the selection process use mention (I have used it with giant pumpkins to good result). My selections have mostly been to save seed within one cultivar that set fruit true to type earliest. On cultivars that produce really big fruit like Aunt Gerties Gold, or Olena's Ukrainian--thats still what I do.

I think if you save seed from the plants that set the biggest fruit on a second set of bloom, you'll end up with a better looking fruit, my first fruit-sets are always prone to cat-facing due to cold night time temperatures up hy're in cow-hampshah'.

I do like a tomato that one slice of 'mater will cover a slice of bread from crust to crust.

Mark how does your selection(s) fare as far as splitting goes?

I don't know how heavy it was, but folks were stopping and asking me how I got pumpkins to ripen in July...LOL It was a Dixie Golden Giant, and one slice of it had to be cut into quarters to fit a hamburger bun.

Mark how does your selection(s) fare as far as splitting goes? i usually cull any ones with signs splitting and shade one that i think may go big in size.