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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\F_3^n$ be the [[vector space|000AYPA9AJ]] of dimension $n$ over the [[field|0ZXZ82FTX2]] $\\F_3$ with three elements.\nA *cap set* is a set $A \\subseteq \\F_3^n$ that contains no three distinct elements $x, y, z$ with $x + y + z = 0$, that is, no three distinct points on a line.\nLet $m_n$ be the number of monomials $x_1^{e_1} \\cdots x_n^{e_n}$ with each $e_i \\in \\set{0, 1, 2}$ and $e_1 + \\cdots + e_n \\le 2n/3$.\nThen every cap set $A \\subseteq \\F_3^n$ satisfies $|A| \\le 3 m_n$.\nIn particular $|A| = o(2.756^n)$ as $n \\to \\infty$, in the sense of [[asymptotic notation|0TGV80E6A4]].\n\n## Remarks\n\nThis bound is due to Ellenberg and Gijswijt, following a method of Croot, Lev and Pach for $(\\Z/4\\Z)^n$.\nThe proof is an instance of the [[linear algebra method|DMYGZ3YZ6V]].\nThe function $T(x, y, z) = 1$ if $x + y + z = 0$ and $0$ otherwise, restricted to $A \\times A \\times A$, is diagonal with $|A|$ nonzero entries, so its slice rank is $|A|$.\nWriting $T(x, y, z) = \\prod_{i} \\left(1 - (x_i + y_i + z_i)^2\\right)$ and expanding, each monomial has degree at most $2n/3$ in one of the three variables, which gives slice rank at most $3 m_n$.\nEarlier bounds, by Fourier methods, were of the form $3^n / n^{1 + c}$.\n\nExample: in $\\F_3^2$ the set $\\set{(0,0), (0,1), (1,0), (1,1)}$ is a cap set of size $4$.\nThe sums of its four triples of distinct points are $(1,1)$, $(1,2)$, $(2,1)$ and $(2,2)$, none of which is $0$.\nCap sets of size larger than $2.2^n$ are known for large $n$, so the base of the exponential lies between $2.2$ and $2.756$.\n\n## Explanations\n\n### L1 human\n\nIn the card game SET, a cap set is a collection of cards that contains no SET.\nThe question is how large such a collection can be when cards have $n$ attributes instead of four.\nThe answer was long known to be a vanishing fraction of all $3^n$ cards, and the 2016 theorem showed it is exponentially smaller: at most about $2.756^n$.\n\n### L2 human\n\nThe cap set problem is the analogue in $\\F_3^n$ of [[Roth's theorem|E7H8NWXESF]] on three-term arithmetic progressions, since three distinct points $x, y, z$ with $x + y + z = 0$ form an arithmetic progression in $\\F_3^n$.\nThe polynomial proof is very short and gives a power saving, which Fourier methods could not.\nIt introduced the slice rank of tensors and led to new bounds on related problems, such as sunflower-free families and matrix multiplication barriers.\nIn the integers, by contrast, there are sets without three-term progressions of size $N^{1 - o(1)}$, so the polynomial method does not transfer directly.\n",
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