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Find the smallest positive integer

{k}

with the following properties

:{}{}{}{}{}

If each positive integer is arbitrarily colored red or blue

{}{}{},

there may be

{}{}{}{}9

distinct red positive integers

x_1,x_2,\cdots ,x_9,

satisfying

x_1+x_2+\cdots +x_8<x_9,

or there are

10{}{}{}{}{}{}

distinct blue positive integers

y_1,y_2,\cdots ,y_{10}

satisfiying

{y_1+y_2+\cdots +y_9<y_{10}}.

Problem Statement