Martingale collars are very important. Greyhounds heads are smaller than their necks, and if they back up they can back right out of a regular collar. A martingale collar tightens if this happens, but doesn't hurt them.

edited to add a better, more descriptive answer

Martingale Collars

A martingale is a type of "limited choke" collar, which is made to slip on over the dog's head (there are no fasteners). Once around the dog's neck, the collar can lie flat, like a buckle collar, or, with a leash attached, it will constrict to the size of the dog's throat. Adjusted correctly, it will not constrict to a size smaller than the dog's throat. In other words, it won't continue to tighten, or "choke" the dog. It does, however, prevent the dog from squirming out of his collar. In it's tightened position, it will not come back off over the dog's head.

We use this type of collar for flyball training. The dog, in his excitement, can not wriggle free of his collar, and the loop gives you and instant "handle" which disappears as soon as you let go. The other flyball collars I have seen, have "built-in" handles, which I would fear could get caught on something, or have a dog put a paw through them while jumping, resulting in potential serious injury.

These collars are also dandy for agility. The loop eliminates the need for a "tag line" (for those that use them), and it allows for control without danger of choking.

We also use these collars for sled dog racing. There is another type of "limited choke" collar used by the majority of sled dog enthusiasts, but it's probably because they haven't discovered the martingale type, yet. Racing rules require that the dogs not be run on choke collars (for safety purposes). But they get so excited to go that they often slip out of the buckle collars. A limited choke collar answers both problems.