When are off-diagonal hypergraph Ramsey numbers polynomial?

A natural open problem in Ramsey theory is to determine those <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> -graphs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the off-diagonal Ramsey number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis upper H comma upper K Subscript n Superscript left-parenthesis 3 right-parenthesis Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">r(H, K_n^{(3)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> grows polynomially with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We make substantial progress on this question by showing that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is tightly connected or has at most two tight components, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis upper H comma upper K Subscript n Superscript left-parenthesis 3 right-parenthesis Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">r(H, K_n^{(3)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> grows polynomially if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is contained in an iterated blowup of an edge.