The current standard of care for patients with heart attacks is to transfuse red blood cells when the hematocrit (red blood cell count) drops below 30 percent. However, there is little scientific basis for this current standard, and recent research has demonstrated that it is safe to allow the hematocrit (red blood cell count) to drop significantly lower in severely ill medical patients and in patients with heart disease undergoing major surgery. The investigators therefore propose this pilot trial to begin to determine whether or not it is safe to apply a more conservative blood transfusion strategy to patients with heart attacks.