Tiling 3-uniform hypergraphs with K_4^3-2e

TL;DR

This paper determines the exact minimum pair-degree threshold needed to guarantee a perfect tiling of 3-uniform hypergraphs with copies of K_4^3-2e, refining previous asymptotic results for large n divisible by 4.

Contribution

It establishes the precise minimum pair-degree conditions for tiling 3-uniform hypergraphs with K_4^3-2e, extending prior asymptotic findings to exact thresholds for large n.

Findings

Exact thresholds for tiling when n/4 is odd or even

Use of absorption technique in hypergraph tiling

Extension of asymptotic results to exact values

Abstract

Let K_4^3-2e denote the hypergraph consisting of two triples on four points. For an integer n, let t(n, K_4^3-2e) denote the smallest integer d so that every 3-uniform hypergraph G of order n with minimum pair-degree \delta_2(G) \geq d contains \floor{n/4} vertex-disjoint copies of K_4^3-2e. K\"uhn and Osthus proved that t(n, K_4^3-2e) = (1 + o(1))n/4 holds for large integers n. Here, we prove the exact counterpart, that for all sufficiently large integers n divisible by 4, t(n, K_4^3-2e) = n/4 when n/4 is odd, and t(n, K_4^3-2e) = n/4+1 when n/4 is even. A main ingredient in our proof is the recent `absorption technique' of R\"odl, Ruci\'nski and Szemer\'edi.