>> assorted mixed heirloom tomatoes,
>> if I grow a tomato that looks exactly like an Arkansas traveler picture, and I really love the taste, would it be safe to assume the seeds will grow an Arkansas traveler?
Since they are all heirlooms, they are all OP. If you can avoid cross-pollination, they will come true to their parent.
Tomatoes are "mostly" self-pollinated, and if you seldom see a bee nosing around the tomato blooms, each plant will produce "mostly" self-fertilized seeds. Even with some bees, "most" blooms will already have been self-pollinated BEFORE the bee got to them, assuming that the vines are shaken or blown around by breezes often.
"Mostly" is something like 80-95% if plants are grown in separate rows, but it sounds like yours will grow right into each other. So if you want to KNOW what you are growing next year, you'll have to identify which heirloom you liked, and buy a packet with that name (or trade for some known seeds).
Or, if you can plant 5 plants and accept that around one of them is likely to be an unplanned hybrid, that is easiest.
Or get fancy. Once you know what plant you want to preserve, make a bag from tulle or buy an organza bag big enough for a bottle of wine. Maybe panty hose or a nylon stocking. Strip any existing fruit and open flowers off the end of one vine to get rid of any possible randomly pollinated blooms. Wait for more blooms to form inside the bag, and vibrate them enough that they pollinate themselves. The organza bag will keep insects away, and tomatoes don't wind-pollinate.
Collect only fruit from inside the bag, and you'll know they were not cross-pollinated.
