On regular 3-wise intersecting families

Ellis and the third author showed, verifying a conjecture of Frankl, that any 3-wise intersecting family of subsets of \{1,2,\dots ,n\} admitting a transitive automorphism group has cardinality o(2^n), while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl, and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any 3-wise intersecting family of subsets of \{1,2,\dots ,n\} that is regular and increasing has cardinality o(2^n).