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  "body": "\n## Statement\n\nLet $A$ be a set of positive integers with positive upper density, that is,\n$$\\limsup_{N \\to \\infty} \\frac{\\abs{A \\cap \\set{1, \\dots, N}}}{N} \u003e 0,$$\nwhere $\\limsup$ is the [[limit superior|0T3RZQXDYD]].\nThen for every integer $k \\ge 1$, the set $A$ contains an arithmetic progression $a, a + d, \\dots, a + (k-1)d$ of length $k$ with $d \\ge 1$.\n\n## Remarks\n\nAn equivalent finite form reads as follows.\nFor every $k \\ge 1$ and $\\delta \u003e 0$ there is $N_0$ such that the following holds for all $N \\ge N_0$.\nEach subset of $\\set{1, \\dots, N}$ with at least $\\delta N$ elements contains an arithmetic progression of length $k$.\nSzemerédi's original proof is combinatorial and introduced the [[Szemerédi regularity lemma|ETMAPNQY7T]].\nOther proofs use ergodic theory (Furstenberg's multiple recurrence theorem), higher-order Fourier analysis with [[Gowers norms|ZBFZGA1J5G]], and hypergraph regularity.\nThe case $k = 3$ is [[Roth's theorem on arithmetic progressions|E7H8NWXESF]].\nThe density hypothesis cannot simply be dropped: the infinite set of powers of $2$ has density zero and contains no three-term arithmetic progression.\nThe primes also have density zero, so the theorem does not apply to them directly; the [[Green–Tao theorem|H8BAHWSEMC]] handles them separately.\n\n## Explanations\n\n### L1 human\n\nIf a set of whole numbers takes up a fixed positive fraction of the numbers, then it contains evenly spaced patterns of every length.\nFor example, it must contain three numbers like $10, 17, 24$, four numbers like $3, 8, 13, 18$, and so on.\nThis holds even if the set looks very irregular.\n\n### L2 human\n\nSzemerédi's theorem is the density version of [[van der Waerden's theorem|V96FRZ2ME1]]: in a finite colouring one colour class has positive upper density, so it contains long progressions.\nIt is a central theorem of [[additive combinatorics|3NM362A2JG]], and the search for good quantitative bounds led to Gowers norms and much of the modern theory.\nThe ergodic proof opened the field of ergodic Ramsey theory, with polynomial and multidimensional extensions.\n",
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