Cut cabbage in half and into quarters. Remove bitter core in center. Cut into small enough chunks that will fit through food processor equipped with smallest grating blade.
Grate cabbage and put in large colander. Rinse thoroughly with cold clean water. Sprinkle liberally with salt and mix. Place colander in sink or on large plate to catch drippings. Allow to sit for approx. 1/2 hour.
Vigorously squeeze juice out of cabbage, one handful at a time. Transfer squeezed cabbage to large bowl.
Grate carrots and onion on same small grater blade in food processor and add to bowl with cabbage. Stir in 2 Tablespoons cider vinegar, 1 Tablespoon celery seed, and sprinkle with pepper.
Add 1/4-1/3 jar mayonnaise (to your preferred taste and moisture level). Mix and adjust seasonings to taste. Will last for 2-3 days in refrigerator.