Turan Problems and Shadows I: Paths and Cycles

TL;DR

This paper determines the maximum edges in large hypergraphs avoiding specific paths and cycles, extending classical extremal combinatorics results with a novel probabilistic shadow sampling approach.

Contribution

It exactly characterizes extremal numbers for paths and cycles in hypergraphs for all sizes and introduces a new method involving random sampling from shadows.

Findings

Exact extremal numbers for paths and cycles in hypergraphs.

Characterization of extremal examples for large n.

Extension of classical conjectures in hypergraph theory.

Abstract

A $k$-path is a hypergraph P_k = e_1,e_2,...,e_k such that |e_i \cap e_j| = 1 if |j - i| = 1 and e_i \cap e_j is empty otherwise. A k-cycle is a hypergraph C_k = e_1,e_2,.. ,e_k obtained from a (k-1)-path e_1,e_2,...,e_{k-1} by adding an edge e_k that shares one vertex with e_1, another vertex with e_{k-1} and is disjoint from the other edges. Let ex_r(n,G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We determine ex_r(n, P_k) and ex_r(n, C_k) exactly for all k \ge 4 and r \ge 3 and $n$ sufficiently large and also characterize the extremal examples. The case k = 3 was settled by Frankl and F\"{u}redi. This work is the next step in a long line of research beginning with conjectures of Erd\H os and S\'os from the early 1970's. In particular, we extend the work (and settle a recent conjecture) of F\"uredi, Jiang and Seiver who solved this problem for P_k when r \ge 4 and of F\"uredi and Jiang who solved it for C_k when r \ge 5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.