= Tight pairs of left-compressed generators
{title2=$m\in G\cap H,\quad |G\cap H|=t\Longrightarrow G\cup H=[m]$}
For a <t-intersecting family> with left-compressed generators supported on $[m]$, a tight pair containing $m$ cannot both miss an earlier coordinate $i$. Replacing $m$ by $i$ in one generator gives a shifted set containing a generator, whose intersection with the other has at most $t-1$ points. Thus the union is $[m]$, and their sizes sum to $m+t$. This identifies precisely which size classes conflict when the largest support coordinate is removed.
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