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Let $I$ be a homogeneous ideal in a polynomial ring $k[x_1,\ldots,x_n],$ $I$ contain a regular sequence $f_1,\ldots,f_n$ such that deg$(f_i)=a_i$ and $a_1\leq\ldots\leq a_n.$ Let $d$ be a non-negative integer. Then there exists a unique lex-plus-power ideal $J$ of the form $J=({x_1}^{a_1},\ldots,{x_n}^{a_n})+L$ where $L$ is a lexicographic ideal generated by monomials of the same degree $d$ with the property $H(I,d)=H(J,d).$ ($H(I,{-})$ is the Hilbert function).

In the paper here "Some cases of the Eisenbud-Green-Harris conjecture" by Giulio Caviglia and Diane Maclagan (page 431), it is written that the following is known:

the Eisenbud-Green-Harris conjecture holds (EGH conjecture is mentioned in the paper) if and only if for any $I,d,J$ defined above, the condition $H(I,d+1)\geq H(J,d+1)$ is also satisfied.

I am unable to prove the above statement. I also could not find any reference for that.

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You may find this appearing many times. See if the sketchy argument in helps (the first 2 pages suffice).

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