2026/07/13 by Balázs Patkós
Mathematics · #math.CO
For a poset P, let e(P) (e^*(P)) denote largest positive integer k such that the union of the k middle layers of 2[n] does not contain a weak (strong) copy of P. Ellis, Ivan, and Leader showed the existence of posets P for which there exists a positive real εP such that La(n,P)≥ (e(P)+εP)\binomn\lfloor n/2 and La^*(n,P)≥ (e^*(P)+εP)\binomn\lfloor n/2 hold, where La(n,P) (La^*(n,P)) denotes the maximum size of a weak (strong) P-free family F⊆ 2[n]. More precisely, they showed that P=Bd are such posets for all d≥ 4, where Bd is the Boolean lattice ordered by inclusion. Tompkins showed that the diamond B2 is also such a poset. We apply his method to settle the case of the last Boolean poset B3. We show that there exists a positive ε such that La^*(n,B3)≥ La(n,B3)≥ La(n,D6)≥ (3+ε)\binomn\lfloor n/2\rfloor, where Ds is the poset on s+2 elements a<b1,…,bs<c. Let ms=min\m:2m-2≥ s\, m^*s=min\m:\binomm\lfloor m/2\rfloor≥ s\. Consider the intervals Im=[2m-1-1,2m-2], I^*m=[\binomm-1\lfloor (m-1)/(2)\rfloor+1,\binomm\lfloor (m)/(2)\rfloor]. It is known that for values s in the major initial parts of Im and Im^*, one has La(n,Ds)=(m+o(1))\binomn\lfloor (n)/(2)\rfloor and La^*(n,Ds)=(m+o(1))\binomn\lfloor (n)/(2)\rfloor. The above equalities do not hold for the largest elements of the intervals, thus there exist sm∈ Im, s^*m∈ I^*m such that for s∈ Im we have La(n,Ds)=(m+o(1))\binomn\lfloor (n)/(2)\rfloor if and only if s<sm and for s∈ I^*m we have La^*(n,Ds)=(m^*+o(1))\binomn\lfloor (n)/(2)\rfloor if and only if s<s^*m. Modifying previous constructions, we obtain upper bounds on sm and s^*m.