Dear Professor Shahriar Shahriari. May I please ask a question? If the corollary weakens the lower bound, what is its benefit? Specifically, when you mentioned making it “look better,” could you elaborate on what this means? I conducted tests with r(9,9) and r(12,12) comparing the lower bounds achievable through this corollary against those obtained using the previous inequality nCr(s,n) < 2^((nCr(n,2)-1)). It appears that the corollary provides considerably lower bounds, and this discrepancy seems to increase with larger values of n. Thank you very much for your insight, and please forgive any of my oversight.
@maodoanminh You are right, the corollary just weakens the original lower bound, and so there is no circumstance that it is actually better. It is just that for, back of envelope calculations, it is easier to use.
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Dear Professor Shahriar Shahriari. May I please ask a question? If the corollary weakens the lower bound, what is its benefit? Specifically, when you mentioned making it “look better,” could you elaborate on what this means? I conducted tests with r(9,9) and r(12,12) comparing the lower bounds achievable through this corollary against those obtained using the previous inequality nCr(s,n) < 2^((nCr(n,2)-1)). It appears that the corollary provides considerably lower bounds, and this discrepancy seems to increase with larger values of n. Thank you very much for your insight, and please forgive any of my oversight.
@maodoanminh You are right, the corollary just weakens the original lower bound, and so there is no circumstance that it is actually better. It is just that for, back of envelope calculations, it is easier to use.