Graham's record-winning tomato came from a Delicious vine variety, so that's obviously a variety to keep in mind. But there are lots of others that are up to the challenge of growing truly giant tomatoes. (Here's more on how to grow bigger, better tomatoes.) Here are some we recommend, and how to grow them big:
Related: 7 Secrets For A High-Yield Vegetable Garden
Tomato varieties to choose from
Most huge tomato varieties are indeterminate (plants that grow and fruit all season) slicer or beefsteak types. Giant Belgium produces sweet fruits that average 2 pounds. Hillbilly is an heirloom beefsteak type with pink-and-yellow bicolored fruits that weigh in at about 1-2 pounds. Big Zac is a hybrid of two heirlooms, and it regularly produces larger 4-6 pound fruits.
Then there are the tomatoes that I like to call the "big beefies": Beefsteak, Italian Giant Beefsteak, Watermelon Beefsteak, Beefmaster, which are all well-known for producing heavy fruits. Check out Tomato Growers Supply Company and Totally Tomatoes for more ideas.
Related: Secrets To Growing Plump Tomatoes
How to grow them huge
It's a lot of work to coax your tomatoes into really packing on the pounds. First of all, you'll need to provide the best basic care: full sun, fluffy soil with plenty of organic matter, lots of space between plants, an inch of water per week, and support like stakes or cages. (Here's more on a basic care growing guide for your tomatoes.)
Get the plants into the ground early and protect them with Wall-O-Waters or cloches. Hold off on mulching until the ground warms up and the plants begin to flower. Fertilize every two weeks with a dilute fish emulsion fertilizer (but be careful not to provide too much nitrogen, or you'll get a lot of foliage and few fruits).
Related: 26 Plants You Should Always Grow Side-by-Side
Once the plants begin to grow, the real training begins. Allow only one stem to develop, and pluck off suckers (the sprouts that form between branches and the main stem) when they are very young. Remove all but two or three fruits from each plant.
It's best to eliminate developing fruits at the top of the vine and leave older fruits at the bottom. Prune off tomatoes that develop farthest from the stem and leave one fruit per cluster. Prevent branches from breaking by supporting the tomatoes with pantyhose slings when they start to get really big (Here's 10 more wacky gardening hacks).
Follow these steps, and wish for some good luck—you're going to need it if you want a shot at knocking down that record!