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Whenever I play the English Opening, I usually try to achieve a c4, d4, e3 pawn structure. However, in this video by STL chess club, the instructor recommends playing d3 (and possibly e4 later) to control the white squares as much as possible.

Which is objectively better? Does it depend on the opponent's play?

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Once you play d4 you are usually leaving the pure English opening and transposing to a queen-pawn opening. Objectively there are positions where it is absolutely better to transpose to a queen-pawn opening but you can certainly build a repertoire around playing the pawn to d3 -- it may just be easier for Black to equalize in some lines.

The formation of c4, d3, e4 with the kings bishop fianchetto'd, knights on e2 and c3 was Botvinnik's baby and is a viable way to play.

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It depends on your opponent's play and your own tastes. Neither are objectively better on their own merits. Yes, c4-d3-e4 does control the light squares, but it grossly weakens the central dark squares (especially d4). Your setup with c4,d4,e3 goes for broad central control, but is rather slow (sometimes allowing Black to strike first with ...d5).

There's nothing wrong with pushing the d-pawn to d4. However, for the sake of speed in some positions, consider just playing c4 and d4 (omitting the move e3).

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One aspect of the English opening is that White retains a lot of options. He believes that c2-c4 will be a useful move in many different situations, so he may play with the idea of keeping his intentions concealed, or with the idea of leaving Black to disclose his intentions first. But it is also possible to play the English having a certain setup, such as the Botwinnik c4, d3, e4 structure, in mind. A common mistake is to do nothing but play "flexible" moves, and leave Black with a free hand to whatever he wants. Keep these considerations in mind as you play through some master games, and imagine whether you would feel comfortable in the situations that arise.

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