Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider an extension of the Monotone Subsequence lemma of Erdos and Szekeres in higher dimensions. Let v1,…,vn ∈ Rd be a sequence of real vectors. For a subset I ⊆ [n] and vector sc ∈ {0,1}d  say that I is sc-free if there are no i < j ∈ I, such that, for every k = 1,…,d, vik < vik if and only if sck = 0.  construct sequences of vectors with the property that the largest sc-free subset is small for every choice of sc. In particular, for d = 2 the largest sc-free subset is O(nf) for all the four possible sc. The smallest possible value remains far from being determined. also consider and resolve a simpler variant of the problem.
['Tibor Szabó', 'Gábor Tardos']
I am writing a research paper on a idea A Multidimensional Generalization of the Erdös–Szekeres Lemma on Monotone Subsequences
[{"from": "human", "value": "I am writing a research paper on a idea A Multidimensional Generalization of the Erd\u00f6s\u2013Szekeres Lemma on Monotone Subsequences"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider an extension of the Monotone Subsequence lemma of Erdos and Szekeres in higher dimensions. Let v1,\u2026,vn \u2208 Rd be a sequence of real vectors. For a subset I \u2286 [n] and vector sc \u2208 {0,1}d  say that I is sc-free if there are no i < j \u2208 I, such that, for every k = 1,\u2026,d, vik < vik if and only if sck = 0.  construct sequences of vectors with the property that the largest sc-free subset is small for every choice of sc. In particular, for d = 2 the largest sc-free subset is O(nf) for all the four possible sc. The smallest possible value remains far from being determined. also consider and resolve a simpler variant of the problem."}]