AN ERDŐS–KO–RADO THEOREM FOR BINARY CODES

<jats:p> We study intersecting families of words from the Erdős–Ko–Rado perspective. When the alphabet size is <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mn>2</mml:mn> </mml:math> <jats:tex-math>$2$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0004972726101452_inline1.png"> <jats:alt-text content-type="machine-generated">2</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> , a maximum intersecting family is not necessarily a star. However, we prove that every maximum <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mn>3</mml:mn> </mml:math> <jats:tex-math>$3$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0004972726101452_inline2.png"> <jats:alt-text content-type="machine-generated">3</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> -wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mn>3</mml:mn> </mml:math> <jats:tex-math>$3$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0004972726101452_inline3.png"> <jats:alt-text content-type="machine-generated">3</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> : maximum intersecting families of words are exactly the stars. </jats:p>