An uniform version of Dvir and Moran's theorem
Abstract
Dvir and Moran proved the following upper bound for the size of a family F of subsets of [n] with Vdim( F F)≤ d. Let d≤ n be integers. Let F be a family of subsets of [n] with Vdim( F F)≤ d. Then \[ | F| 2Σk=0 d/2 nk. \] Our main result is the following uniform version of Dvir and Moran's result. Let d≤ n be integers. Let F be an uniform family of subsets of [n] with Vdim( F F)≤ d. Then \[ | F| 2 n d/2 . \] Denote by vF∈ \0,1\n the characteristic vector of a set F ⊂eq [n]. Our proof is based on the following uniform version of Croot-Lev-Pach Lemma: Let 0≤ d≤ n be integers. Let H be a k-uniform family of subsets of [n]. Let F be a field. Suppose that there exists a polynomial P(x1, … ,xn,y1, … ,yn)∈ F[x1, … ,xn,y1, … ,yn] with deg(P)≤ d such that P(vF,vF) 0 for each F∈ H and P(vF,vG)= 0 for each F G∈ H. Then \[ | H| 2 n d/2 . \]
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