The Borzoi is a Russian hunting dog that was bred by the Tsars and nobility of Russia to be the ultimate runner and chaser. He is descended from ancient sighthounds like the Russian smooth-faced bearhound, Russian sheepdogs, and Southern coursing hounds. All of these dogs had one thing in common: they were prominent figures in the Russian tsars' hunting parties. The result of all of this breeding was the Russian Wolfhound or Borzoi, who is one of the faster and most intelligent sighthounds today. But what does all of this have to do with cats?
Think about the purpose of hunting dogs for a moment: what are they trained to do? And consider not only what their owner trains them to do, but what they are bred to do. They are bred and trained to chase small animals, right? These dogs, like the Borzoi, are specifically bred from large breed dogs to create an intelligent, fast, and chase-prone animal that loves to run after what it considers prey. And while this is not a problem for people who do not have small animals in their home with their Borzoi, it can be a serious problem in homes with cats.
The Borzoi is a coursing or chasing dog. When he sees an animal running from him, he will be inclined to chase after it. And while the Borzoi were not bred to kill their prey when caught - generally this task was left for the human hunters, only requiring the Borzoi dogs to "capture" and hold the prey by the throat until the people arrived - they will chase and possibly hurt an animal they are pursuing.
By nature, the Borzoi will chase, and by nature, a cat will run when chased. This is quite obviously a bad combination for all involved since the dog will be relentless in his chase and will easily overtake the cat. This can be harmful to the cat in a number of ways, not the least of which is the accidental harm that can come from the Borzoi actually catching the cat. If the dog follows his instinct and grasps the cat by the throat, the cat could easily end up with a broken neck. But there are other things to worry about, too. Even if the Borzoi doesn't catch the cat, the cat may be permanently behaviorally affected by the chase, and could exhibit aggressive tendencies towards the dog. The cat could also become so afraid of the dog so that it will hide or become skittish and unfriendly.
Obviously, none of these options are good for either the dog or the cat involved. But this is not to say that Borzoi will always harm a cat. When a Borzoi is raised with a cat, he may be trained not to chase the cat. Another option is to choose a male rather than a female Borzoi. Females have a stronger tendency to chase than males do. And, of course, the personality of the cat will also ultimately determine the outcome of the relationship between the Borzoi and the cat.