Set systems and families of permutations with small traces

Let F be a family of permutations on [ n ] = { 1 , … , n } and let Y = { y 1 , … , y m } ⊆ [ n ] , with y 1 < y 2 < ⋯ < y m . The restriction of a permutation σ on [ n ] to Y is the permutation σ | Y on [ m ] such that σ | Y ( i ) < σ | Y ( j ) if and only if σ ( y i ) < σ ( y j ) ; the restriction of F to Y is F | Y = { σ | Y ∣ σ ∈ F } . Marcus and Tardos proved the well-known conjecture of Stanley and Wilf that for any permutation τ on [ m ] there is a constant c such that if no permutation in F admits τ as a restriction then F has size O ( c n ) . In the same vein, Raz proved that there is a constant C such that if the restriction of F to any triple has size at most 5 (regardless of what these restrictions are) then F has size at most C n . In this paper, we consider the following natural extension of Raz’s question: assuming that the restriction of F to any m -element subset in [ n ] has size at most k , how large can F be? st investigate a similar question for set systems. A set system on X is a collection of subsets of X and the trace of a set system R on a subset Y ⊆ X is the collection R | Y = { e ∩ Y ∣ e ∈ R } . For finite X , we show that if for any subset Y ⊂ X of size b the size of R | Y is smaller than 2 i ( b − i + 1 ) for some integer i then R consists of O ( | X | i ) sets. This generalizes Sauer’s Lemma on the size of set systems with bounded VC-dimension. We show that in certain situations, bounding the size of R knowing the size of its restriction on all subsets of small size is equivalent to Dirac-type problems in extremal graph theory. In particular, this yields bounds with non-integer exponents on the size of set systems satisfying certain trace conditions. n map a family F of permutations on [ n ] to a set system R on the pairs of [ n ] by associating each permutation to its set of inversions. Conditions on the number of restrictions of F thus become conditions on the size of traces of R . Our generalization of Sauer’s Lemma and bounds on certain Dirac-type problems then yield a delineation, in the ( m , k ) -domain, of the main growth rates of F as a function of n .