Compressions and Probably Intersecting Families
Paul A. Russell

TL;DR
This paper investigates the structure of large intersecting families of sets, proving a conjecture about optimal families for maximizing intersection probability after random sampling, using novel compression techniques.
Contribution
It introduces a new compression method to identify optimal families of sets, confirming a conjecture for certain sizes and exploring intermediate cases.
Findings
Families of the form [n]^( extgreater r) are optimal for specific sizes.
Optimal families maximize the number of intersecting subfamilies of all orders.
Standard compression techniques are inadequate; a new method is developed.
Abstract
A family X of sets is said to be intersecting if any two members of X have non-empty intersection. It is a well-known and simple fact that an intersecting family of subsets of [n]={1,2,...,n} can contain at most 2^(n-1) sets. Katona, Katona and Katona ask the following question. Suppose instead a family X of subsets of [n] satisfies |X|=2^(n-1)+i for some fixed i>0. Create a new family X_p by choosing each member of X independently with some fixed probability p. How do we choose X to maximize the probability that X_p is intersecting? They conjecture that there is a nested sequence of optimal families for i=1, 2, ..., 2^(n-1). In this paper, we show that the families [n]^(\ge r)={A\subset[n]:|A|\ge r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
