Exact Results on Traces of Sets

For non-negative integers n, m, a and b, we write (n,m)→(a,b) if for every family F⊆ 2^[n] with |F|≥ m there is an a-element set T⊆[n] such that |F_|T|≥ b, where F_|T={F∩T:F∈F}. A longstanding problem in extremal set theory asks to determine m(s)=lim_{n→+∞} m(n,s)/n, where m(n,s) denotes the maximum integer m such that (n,m)→(n-1,m-s) holds for non-negatives n and s. In this paper, we establish the exact value of m(2^{d-1}-c) for all 1≤ c≤ d whenever d≥ 50, thereby solving an open problem posed by Piga and Schülke. To be precise, we show that m(n,2^{d-1}-c) = { (2^{d - c)/d * n for 1≤ c≤ d-1 and d does not divide n, (2}d - d - 0.5)/d * n for c=d and 2d does not divide n } holds for d≥ 50. Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that m(11)=5.3.