The Random Walk Method for Intersecting Families

Abstract

ABSTRACT. Let m(n, k, r,t) be the maximum size of F ⊂ [n] k satisfying |F 1 ∩· · ·∩F r | ≥ t for all F 1 , . . . , F r ∈ F . We report some known results about m (n, k, r,t). The random walk method introduced by Frankl is a strong tool to investigate m (n, k, r,t). Using a concrete example, we explain the method and how to use it.

A product version of the Erdős–Ko–Rado theorem

“…Then we will deduce Theorem 6 from Theorem 7 in Section 4, where the key idea is simply that the binomial distribution B(n, p) is concentrated around pn. The other theorems follow from Theorem 6, Theorem 7, and some results from [26,27]. We include these easy proofs in Appendix A.…”

“…The problem of determining m(n, k, r, t) goes back to Erdős, Ko and Rado [4], and is still wide open. All known results, e.g., [1,[4][5][6][7]12,[22][23][24][25][26][27]29], suggest m(n, k, r, t) = max A (n, k, r, t) .…”

On cross t-intersecting families of sets

“…For the proof of our results, we will use the random walk method developed by Frankl in [6,7], and a technique translating results about p-weight version to k-uniform version, cf. [13]. We will also include stability type results, see Theorems 5 and 6 at the ends of the following sections.…”

“…Suppose that F and G are cross t-intersecting. If s ∈ Z and y 0 x 0 + s, then the random walk (starting from (x 0 , y 0 )) hits the line y = x + s with probability α s+x 0 −y 0 , where α = p/q (see [7,13]). Applying this to the case x 0 = y 0 = 0 and s = λ(F ), we have w p (F ) α λ(F ) , because w p (F ) is bounded from above by the probability that the random walk (starting from the origin) hits the line y = x + λ(F ) within the first n steps.…”

Invitation to intersection problems for finite sets

“…Then one can show that for each G ∈ G the walk corresponding to G hits a line y = (r − 1)x + t. This enables us to bound the size of G by counting the number of all n-step lattice walks that hit the line, or equivalently, by the probability that a random walk starting from the origin, with one step up or to the right, hits the line. See [56,70,147] for more details.…”