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| ==A weak multidimensional DHJ(k) implies DHJ(k)==
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| It is also true that a weak multidimensional DHJ(k) implies DHJ(k). We will show that the following statement is equivalent to DHJ(k):
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| “There is a constant, c < 1 that for every d there is an n that any c-dense subset of <math> [k]^n</math> contains a d-dimensional subspace.”
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| We should show that the statement above implies DHJ(k). As before, write <math> [k]^n</math> as <math> [k]^r\times[k]^s </math> , where s is much bigger than r. For each <math> y\in [k]^s</math> , define <math> \mathcal{A}_y</math> to be <math> \{x\in[k]^r:(x,y)\in\mathcal{A}\}</math> . Let Y denote the set of <math> y\in [k]^s</math> such that <math>\mathcal{A}_y</math> is empty. Suppose that <math> \mathcal{A} </math> is large, line-free, and its density is <math> \delta =\Delta-\epsilon</math> where <math> \Delta</math> is the limit of density of line-free sets and <math> \epsilon < (1-c)\Delta</math> . We can also suppose that no <math> \mathcal{A}_y</math> has density much larger than <math> \Delta</math> as that would guarantee a combinatorial line. But then the density of Y is at most 1-c, so there is a c-dense set, <math> Z=[k]^s-Y</math>, such that any element is a tail of some elements of <math> \mathcal{A}</math> . For every <math> y \in Z</math> choose an <math> x\in [k]^r:(x,y)\in\mathcal{A}</math> . This x will be the colour of y. It gives a <math> [k]^r</math> colouring on Z. By the initial condition Z contains arbitrary large subspaces, so by HJ(k) we get a line in <math> \mathcal{A}</math> .
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