The sturdy striped or patterned fabric traditionally used to make the outer covering of mattresses and pillows.
From bedtick + -ing forming an abstract noun/material noun. The fabric became known by this name because it was specifically manufactured for making bed ticks, and 'bedticking' refers to the fabric type itself.
Bedticking's characteristic stripes were originally functional—the stripe patterns helped identify different quality grades and manufacturers, so experienced shoppers could tell bed quality at a glance.
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