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Simonovits and Sos proved $t(N)=O(N^2)$ and conjectured $\\binom{N}{2}+1$ is best possible; Szabo disproved this by a construction giving $t(N) \\geq \\binom{N}{2}+1+\\lfloor(N-1)/4\\rfloor$, proved the asymptotics $t(N)=N^2/2+O(N^{5/3}(\\log N)^3)$, and asked whether $t(N)=\\binom{N}{2}+O(N)$ and whether some element lies in all sets of any extremal family (the kernel question). We determine $t(N)$ exactly for all $3 \\leq N \\leq 12$ by exhaustive computation: in this entire range Szabo's lower bound is exact, and we conjecture that $t(N)=\\binom{N}{2}+1+\\lfloor(N-1)/4\\rfloor$ for every $N$. Towards the matching upper bound we prove, for every $N$, that Szabo's bound is the exact maximum over all families with a common element (starred fa","title":"Exact values and exact upper bounds for families of integers with arithmetic progression intersections (Erd\\H{o}s Problem #272)","url":"https://arxiv.org/abs/2607.23004","vendor":"arxiv_cs_ai"},"summary":"arXiv:2607.23004v1 Announce Type: cross \nAbstract: Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\\dots,A_t \\subseteq \\{1,\\dots,N\\}$ such that $A_i \\cap A_j$ is a nonempty arithmetic progression for all $i \\neq j$ (Erdos Problem #272). Simonovits and Sos proved $t(N)=O(N^2)$ and conjectured $\\binom{N}{2}+1$ is best possible; Szabo disproved this by a construction giving $t(N) \\geq \\binom{N}{2}+1+\\lfloor(N-1)/4\\rfloor$, proved the asymptotics $t(N)=N^2/2+O(N^{5/3}(\\log N)^3)$, and asked whether $t(N)=\\binom{N}{2}+O(N)$ and whether some element lies in all sets of any extremal family (the kernel question). We determine $t(N)$ exactly for all $3 \\leq N \\leq 12$ by exhaustive computation: in this entire range Szabo's lower bound is exact, and we conjecture that $t(N)=\\binom{N}{2}+1+\\lfloor(N-1)/4\\rfloor$ for every $N$. Towards the matching upper bound we prove, for every $N$, that Szabo's bound is the exact maximum over all families with a common element (starred fa","title":"Exact values and exact upper bounds for families of integers with arithmetic progression intersections (Erd\\H{o}s Problem #272)","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-07-28T04:43:08Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2607.23004"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:622d26c4b27d362018c5af5d2630d0b8714ad9820214d2f89bbe9f0423398d6d84accb2283990b362ddf0170228f329928da1f4a33a8afd4c08ea554b7be9808","signer":"crovia.substrate","subject":{"observed_at":"2026-07-28T04:43:08Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2607.23004"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"13606335163918ce28963977aec2b101ef77a13d3b2502ec705f7be3b4664d79","leaf_index":360564,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"4bba9d126a5c3f027a613f6d77885378f1d0789e28bd81832474da8cf7512638","side":"right"},{"sibling":"0bababd2819f7c8f4cfe84a6fae742920419241e31531c0eff4e154d224d6148","side":"right"},{"sibling":"1dceaa5105644f41d37a8adeaeb9bea43d7c00e6e60039e64e4b5e048583724f","side":"left"},{"sibling":"8e11a27cda34ad0caa384e8b3257b559c466f12ab597dae30e305a694f5a208e","side":"right"},{"sibling":"86c8dc0d477488431c3de5f1b53b0398d338f00f6c29ebe77548f3becdad6eb0","side":"left"},{"sibling":"683739b49831eb9d8ed7ad240c7843a793d04a14739fa08c2deb614e19a4e9d8","side":"left"},{"sibling":"a75ea3fe312c1d9ab3776bfe70acd45752855705bf7bb0ab8e19b5f21d865d19","side":"left"},{"sibling":"7da15787b28ee42d6f6ffa689ccb25f3c7958b4ad112266a2a4acad4347e0386","side":"right"},{"sibling":"eb79f1d599c07786d2268140481af8b617999ecc5688c6283fe58e5e6f3a2640","side":"right"},{"sibling":"fe03c6d0b083c4097049d7fc6d7d08dfdb05d9c198b2786336689712d66eabf0","side":"right"},{"sibling":"590ea76fdfc1b9e8072055c378be3182f916709e8d06bd955232e650a7188b86","side":"right"},{"sibling":"d92781c59301ffd5bfb0bad75d9fdf6d73879f715518149d383f7362213e0daf","side":"right"},{"sibling":"f315303d4402b57497416d48eb4c4bb50405b40862d41c7caf318cb3d29c5237","side":"right"},{"sibling":"36973eb5f586cd67e0c0dc055dd87e734aa544c35d4400d2c9f932f8ef8fb27f","side":"right"},{"sibling":"e39f7900355489c4718b21cc2d3d06382e1d2f22b864d10be9ebc9d498b279a1","side":"right"},{"sibling":"1f9a970b25dd938c98cabc9e0a55c5a6f46292b9fa37cc89d90ef0cbb1e05a8c","side":"left"},{"sibling":"77025bcb374a7ad74f520643e20a8ae1205a7ee78507b0beb93117760f1c29d3","side":"left"},{"sibling":"3b50864499c874394ea0928567747666eaf59b01380e46cd52164ec5acec0f71","side":"right"},{"sibling":"1cecb7f447febd025aac272837c80de218aecc6485d2395a509b2a1f1b9c746e","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":361008,"merkle_root":"3065e8369ea437c06beba806dc4e4bb159979adeb21fe632242c1906a7204647","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260728T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-07-28T05:38:48Z","sig_algorithm":"ed25519","signature":"9141644407577a82611c1579110f667de2d46dc6b93c6322edf26f4c3056ea99f0e56502853908e30d87c38bcf95eb6e0ab5130525aa51505bd6f61938120609","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_f912b1c77b4a67f6689c610a52878b12fd1bd2db83b485479167e27f2b75359d"}}