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 | FT | b, where FT=\ F ∩ T : F ∈ F \. A longstanding problem in extremal set theory asks to determine m(s)=limn→ +∞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(2ᵈ⁻¹-c) for all 1 c d whenever d 50, thereby solving an open problem posed by Piga and Sch\"{u}lke. To be precise, we show that m(n,2ᵈ⁻¹-c)=2ᵈ-cdn for 1≤ c≤ d-1 and d n, and m(n,2ᵈ⁻¹-d)=2ᵈ-d-0.5dn for 2d n holds for d≥ 50. Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that $m(11)=5.3$.