Given
$\alpha \gt 0$
and an integer
$\ell \geq 5$
, we prove that every sufficiently large
$3$
-uniform hypergraph
$H$
on
$n$
vertices in which every two vertices are contained in at least
$\alpha n$
edges contains a copy of
$C_\ell ^{-}$
, a tight cycle on
$\ell$
vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.
For integers 𝑘 ⩾ 2 and 𝑁 ⩾ 2𝑘 + 1 there are 𝑘!2 𝑘 canonical orderings of the edges of the complete 𝑘-uniform hypergraph with vertex set [𝑁] = {1, 2, … , 𝑁}. These are exactly the orderings with the property that any two subsets 𝐴, 𝐵 ⊆ [𝑁] of the same size induce isomorphic suborderings. We study the associated canonisation problem to estimate, given 𝑘 and 𝑛, the least integer 𝑁 such that no matter how the 𝑘-subsets of [𝑁] are ordered there always exists an 𝑛-element set 𝑋 ⊆ [𝑁] whose 𝑘-subsets are ordered canonically. For fixed 𝑘 we prove lower and upper bounds on these numbers that are 𝑘 times iterated exponential in a polynomial of 𝑛.M S C 2 0 2 0 05D10 (primary), 05C55 (secondary)
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