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• W2008066809 abstract "This paper establishes a connection between a certain class of Ramsey numbers for graphs and the class of symmetric block designs admitting a polarity. The main case considered here relates the projective planes over Galois fields to the Ramsey numbers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R left-parenthesis upper C 4 comma upper K Subscript 1 comma n Baseline right-parenthesis equals f left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>R({C_4},{K_{1,n}}) = f(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that, for every prime power <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q comma f left-parenthesis q squared plus 1 right-parenthesis equals q squared plus q plus 2> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>q,f({q^2} + 1) = {q^2} + q + 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis q squared right-parenthesis equals q squared plus q plus 1> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>f({q^2}) = {q^2} + q + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2008066809 date "1975-01-01" @default.
• W2008066809 modified "2023-10-16" @default.
• W2008066809 title "Ramsey graphs and block designs. I" @default.
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