On regular 3-wise intersecting families

<p>Ellis and the third author showed, verifying a conjecture of Frankl, that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-wise intersecting family of subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="StartSet 1 comma 2 comma ellipsis comma n EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<! -- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{1,2,\dots ,n\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admitting a transitive automorphism group has cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="o left-parenthesis 2 Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">o(2^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-wise intersecting family of subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="StartSet 1 comma 2 comma ellipsis comma n EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<! -- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{1,2,\dots ,n\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is regular and increasing has cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="o left-parenthesis 2 Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">o(2^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. </p>