在離散幾何中有個有趣的問題是尋找最大k-距離集。 即使看似簡單的最大平面7-距離集也還是未知的。

此篇論文我們給出部分結論。 Erdös and Fishburn [1] 給出了16個點的平面7距離集， 但不知道是否是最大的。 我們將17個點的平面7-距離集以X_D的基數做分類， 這個數會介於2到17之間。 我們照著 Wei [2] 的思路研究17個點的平面7-距離集。

我們證明9-13以外是不可能的， 但9-13的部分只能給出部分結論。

It is known that obtaining maximum k-distance sets has been an interesting problem in discrete geometry. Even a seemingly not-difficult problem like the maximum cardinality of 7-distance set in R^2 is yet to be found.
In this thesis we provide some partial results for this problem. Erdös and Fishburn [1] showed the 16-point 7-distance sets, but did not prove that 16 is the maximum. We observe whether there is any 17-point 7-distance set in R^2 based on the cardinality of X_D, where 2≤|X_D |≤17. We follow the method used in Wei [2] for this observation.
We can only provide partial results for 9≤|X_D |≤13, but for the other parts, we prove that there is no 17-point 7-distance set with that value of |X_D |.

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[2] O. R. Musin and H. Nozaki. “Bounds on Three- and Higher-distance Sets”. In: European J. Combin. 32 (2011), pp. 1182–1190.
[3] P. Erdos and P. Fishburn. “Maximum Planar Sets that Determines k Distances”. In: Discrete Mathematics 160 (1996), pp. 115–125.
[4] X. Wei. “A Proof of Erdös-Fishburn’s Conjecture for g(6) = 13”. In: The Electronic Journal of Combinatorics 19(4) (2012).
[5] M. Shinohara. “Uniqueness of Maximum Planar Five-distance Sets”. In: Discrete Mathematics 308 (2008), pp. 3048–3055.
[6] X. Wei. “Classification of Eleven-point Five-distance Sets in the Plane”. In: Ars Combinatoria 102 (2011), pp. 505–515.
[7] E. Altman. “On a Problem of P. Erdös”. In: American Mathematical Monthly 70 (1963), pp. 148–157.