## Abstract

A balanced pattern of order 2d is an element P ∈ {+, −}

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 ad-balanced pattern with d < c log log n then |

^{2d}, whereboth signs appear d times. Two sets A, B ⊂ [n] form a P-pattern,which we denote by pat(A, B) = P, if A△B = {j_{1}, . . . , j_{2d}} with 1 ≤ j_{1}< · · · < j_{2d}≤ n and {i ∈ [2d] : P_{i}= +} = {i ∈ [2d] : j_{i}∈ A \ B}.We say*A*⊂*P*[n] is P-free if pat(A, B) ≠ P for all A, B ∈*A*. Weconsider 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 ad-balanced pattern with d < c log log n then |

*A*|= o(2^{n}). Wethen 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), thisis tight.Original language | English |
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Pages (from-to) | 183-196 |

Number of pages | 14 |

Journal | European Journal of Combinatorics |

Volume | 62 |

Early online date | 16 Jan 2017 |

DOIs | |

Publication status | Published - 1 May 2017 |

## Keywords

- Extremal set theory
- patterns