Set families with a forbidden pattern

A balanced pattern of order 2d is an element P ∈ {+, −}2d , where both signs appear d times. Two sets A, B ⊂ [n] form a P-pattern, which we denote by pat(A, B) = P, if A△B = {j1, . . . , j2d} with 1 ≤ j1 < · · · < j2d ≤ n and {i ∈ [2d] : Pi = +} = {i ∈ [2d] : ji ∈ A \ B}. We say A ⊂ P [n] is P-free if pat(A, B) ̸= P for all A, B ∈ A. We consider the following extremal question: how large can a family A ⊂ P [n] be if A is P-free? We prove a number of results on the sizes of such families. In particular, we show that for some fixed c > 0, if P is a d-balanced pattern with d < c log log n then | A |= o(2 n ). We then give stronger bounds in the cases when (i) P consists of d+ signs, followed by d− signs and (ii) P consists of alternating signs. In both cases, if d = o( √ n)then | A |= o(2 n ). In the case of (i), this is tight

A note on tilted Sperner families with patterns

Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\mathcal{F} \subset \mathcal{P}([n])$ a \textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \in \mathcal{F}$ with: $$(i) \ \ p|F \setminus G|=q|G \setminus F|, \ \textrm{and}$$ $$(ii) \ f > g \ \textrm{for all} \ f \in F \setminus G \ \textrm{and} \ g \in G \setminus F.$$ Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\sqrt{\log n}}\ \frac{2^n}{\sqrt{n}}).$$ We improve and generalize this result, and prove that the cardinality of every ($p,q$)-tilted Sperner family with patterns on [$n$] is $$O(\sqrt{\log n} \ \frac{2^n}{\sqrt{n}}).$$Comment: 8 page

Set families with a forbidden pattern

A balanced pattern of order 2d is an element P ∈ {+, −}2d , where both signs appear d times. Two sets A, B ⊂ [n] form a P-pattern, which we denote by pat(A, B) = P, if A△B = {j1, . . . , j2d} with 1 ≤ j1 < · · · < j2d ≤ n and {i ∈ [2d] : Pi = +} = {i ∈ [2d] : ji ∈ A \ B}. We say A ⊂ P [n] is P-free if pat(A, B) ̸= P for all A, B ∈ A. We consider the following extremal question: how large can a family A ⊂ P [n] be if A is P-free? We prove a number of results on the sizes of such families. In particular, we show that for some fixed c > 0, if P is a d-balanced pattern with d < c log log n then | A |= o(2 n ). We then give stronger bounds in the cases when (i) P consists of d+ signs, followed by d− signs and (ii) P consists of alternating signs. In both cases, if d = o( √ n)then | A |= o(2 n ). In the case of (i), this is tight

Set families with a forbidden pattern

A balanced pattern of order 2d is an element P ∈ {+, −}2d , where both signs appear d times. Two sets A, B ⊂ [n] form a P-pattern, which we denote by pat(A, B) = P, if A△B = {j1, . . . , j2d} with 1 ≤ j1 &lt; · · · &lt; j2d ≤ n and {i ∈ [2d] : Pi = +} = {i ∈ [2d] : ji ∈ A \ B}. We say A ⊂ P [n] is P-free if pat(A, B) ̸= P for all A, B ∈ A. We consider the following extremal question: how large can a family A ⊂ P [n] be if A is P-free? We prove a number of results on the sizes of such families. In particular, we show that for some fixed c &gt; 0, if P is a d-balanced pattern with d &lt; c log log n then | A |= o(2 n ). We then give stronger bounds in the cases when (i) P consists of d+ signs, followed by d− signs and (ii) P consists of alternating signs. In both cases, if d = o( √ n)then | A |= o(2 n ). In the case of (i), this is tight