A "meta-study" (i.e. a study of studies) was made around 1998/1999 by the Public Health Authority of East Anglia (UK).
They looked at all the studies that seemed to suggest that heavy meat eaters had a higher incidence of colonic cancer.
What they found, in fact, was that it was impossible to conclude that high meat consumption increased risks of that cancer. But rather that deficiency in fruit and vegetable consumption could definitely be blamed. (Heavy meat eaters are more likely to leave more veggies out of their diet, hence the mistaken conclusion of the earlier studies.)
In other words, a lot of meat in itself probably won't increase risks of cancer (or the effects are so small that nobody has measured them yet). However, make sure you eat plenty and a good variety of vegetables.
(Every kind of vegetables is good, not just greeny and fibrous ones, served in just about any shape or form: including tomato, onion, cauliflower, carrot, garlic, celeriac, parsnip, Jerusalem artichoke, turnip, sweed, white asparagus, pumpkin and all gourds, endive, sweet pepper red, yellow and orange, as well as all the more obvious greeny vegetables, and all fresh herbs (parsley, basil, celantro-coriander, mint, lemonbaum, thyme, savory, sage, chervil), and all legumes and lentils, etc. There is so much to choose from: no reason to limit yourself to broccoli or green peas!)