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Here we improve it to $f(k)<2^{ck^2}$ for some constant $c>0$.\n  This is a special case of the following result on the multi-coloured forbidden configurations at 2 colours. Let $r$ be given. Then there exists a constant $c_r$ so that a matrix with entries drawn from $\\{0,1,...,r-1\\}$ with at least $2^{c_rk^2}$ different columns will have a $k\\times k$ submatr"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1409.4123","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-15T00:08:38Z","cross_cats_sorted":[],"title_canon_sha256":"6e04374fa2292d0133b81122f76814f658b1c8c931efb6b77541293d4fb63461","abstract_canon_sha256":"1b50f33bd5d8a3ef19af8de97567183e0d0b47bff175c3e20a7d245ac3797f4f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:40.934608Z","signature_b64":"zjZWxQPuF7wR9fqp1LLh6Cug6rfCviodCFexFnAzXaIA/Re0iui2InXjdxV0UHpBNxpEp0XgD3gnJndEnw6TAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a83cd34cab7c8df7d8f4e293afa395757ab05873ef539951aa7e67dafb4339cd","last_reissued_at":"2026-05-18T02:41:40.933944Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:40.933944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unavoidable Multicoloured Families of Configurations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Richard P. 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