{"id":"f8a67f60-bd02-4945-a686-1bde839155cc","arxiv_id":"2506.21993","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For large n, the maximum product is attained only by a pair both equal to the same full star of sets containing a fixed t-set, and all non-trivial extremal configurations are characterized.","lead":"Two families of k-element sets are called s-almost cross-t-intersecting when each set misses the intersection threshold on at most s partners in the other family. This paper determines which such pairs maximize the product of the two family sizes, and describes the extremal pairs explicitly, including a stability version.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"External alternating-sequence bound [19, Thm 6] and unverified Section 7 constants are the load-bearing dependencies; no internal flaw found, but both should be checked.","rationale":"I read the proof structure in detail. Lemma 2.1 and Lemma 2.2 are internally consistent under the stated n-ranges; the chain argument and the geometric-sum bounds check out. Lemma 2.4's construction of the alternating sequences is valid: F_i are distinct because each step removes F_i, and the conditions (a) and (b) follow from the choice rules. The only truly external input is the bound from Oum and Wee, which the paper cites without proof. The Section 7 inequalities are long but I traced several (Lemma 7.2(i), Lemma 7.4(i,iii), Lemma 7.5(ii)) and found the algebra coherent, with constants that are tight but satisfied under the stated thresholds. The extremal constructions in Examples 4.1, 5.1, and 5.2 are valid s-almost cross-t-intersecting families, and the covering-number case analyses in Theorems 1.2 and 1.3 appear complete. The reader's weakest-assumption assessment matches mine: the external bound and the unverified Section 7 arithmetic are the load-bearing points. Since no actual error was identified, I do not recommend changing the ACCEPT verdict, but I recommend that referees verify the external theorem and run a CAS check on Section 7.","tokens_in":23743,"tokens_out":53744,"duration_ms":482770,"concrete_test":"Run a backtracking search over alternating sequences of distinct k-subsets of [n] for t=2,k=3,n=6, enforcing conditions (a) |F_i∩G_i|<t and (b) |F_i∩G_j|≥t for all j<i, and compare the maximum attainable length with the claimed bound C(2k−2t+2,k−t+1)=C(4,2)=6. If a sequence of length 7 exists, Lemma 2.4's bound is false and Theorem 1.1 needs repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1's proof forces (τ_t(F),τ_t(G))=(t,t) by two routes. The route τ_t≥k+1 relies entirely on Lemma 2.4, whose only substantive input is the external bound m≤C(2k−2t+2,k−t+1) from [19, Theorem 6], applied to the greedy sequences F_i∈F, G_i∈G with (a) |F_i∩G_i|<t and (b) |F_i∩G_j|≥t for j<i. The paper does not reproduce this theorem or check that its hypotheses (e.g., whether F_i and G_i must be distinct, or whether extra restrictions on n are needed) match the constructed sequences. If the bound is misquoted or carries an n-dependent condition, the contradiction |F||G|<C(n−t,k−t)^2 in the τ_t≥k+1 case fails. The other route (both covering numbers ≤k) depends on Lemma 7.2(i), and the Section 7 estimates (Lemmas 7.4, 7.5) are long binomial manipulations with no CAS verification; a single strict-inequality error would break the product upper bounds. I found no internal inconsistency, but these are exactly where the central claim is most exposed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies pairs of families F,G of k-subsets of an n-set that are s-almost cross-t-intersecting, meaning that each member of either family is t-disjoint from at most s members of the other family. The main results are as follows. Theorem 1.1 states that, for k ≥ t+1 and n ≥ (t+1)(2(k−t+1)^2+7s), the maximum product |F||G| is attained exactly by two copies of the full t-star H1([n],W;k). Theorems 1.2 and 1.3 classify the extremal pairs that are not cross-t-intersecting: one family is a star with a specified slice removed, the other is that star together with a small block, with the precise parameters depending on s, t, and k; the case k=t+1 is treated separately. Corollaries 6.2 and 6.3 use the non-cross-t-intersecting classification together with the external result [5, Theorem 1.2] to give a stability version under the condition that the common intersection of all members of F∪G has size less than t. The proofs are built on t-covering numbers, several auxiliary bounds on subfamilies, and a long collection of binomial inequalities assembled in Section 7.","tokens_in":23936,"tokens_out":6278,"duration_ms":70390,"significance":"If correct, the paper supplies a natural product-maximum analogue of the Erdős–Ko–Rado and Hilton–Milner line of results for a locally relaxed cross-intersection condition, and it does so in full structural detail rather than only with an extremal number. The extremal constructions are explicit and their product formulas match the upper bounds derived in the proofs, which is a real strength. The t-covering-number framework is appropriate, and the separation of the k=t+1 case in Theorem 1.3 is well matched to the different behaviour of the examples. The proofs are not machine-checked, but the chains of inequalities in Section 7 are written out in enough detail that I found no internal contradiction; the main risk is the dependence on one external theorem and on several delicate rational estimates, as detailed below. If those are verified, the paper gives a complete and convincing characterization.","major_comments":[{"comment":"Lemma 2.4 is load-bearing for Theorem 1.1: the proof rules out the case τ_t(F) ≥ k+1 by applying the alternating-sequence bound m ≤ C(2k−2t+2, k−t+1) from [19, Theorem 6] to sequences F_i, G_i satisfying conditions (a) and (b). The manuscript does not state the exact formulation of [19, Theorem 6] or verify that its hypotheses are satisfied by the constructed sequences, in particular whether it requires the F_i and G_i to be distinct or imposes any condition on n beyond the ones already present. I request that the authors state the theorem explicitly and check those hypotheses; if [19, Theorem 6] carries any hidden n-dependence or distinctness condition, the contradiction in the τ_t ≥ k+1 case would fail.","section":"Lemma 2.4 and Section 3"},{"comment":"The central product upper bounds ultimately rest on a sequence of delicate rational inequalities and monotonicity claims, including the bounds 7225/14112, 69803/112896, 2119/14112 in Lemma 7.4, the monotonicity of f3 in Lemma 7.5, and the comparison g1 < g4 in Lemma 7.6. I did not find a specific arithmetic error, but these estimates are described as routine computations and are load-bearing for Theorems 1.1, 1.2, and 1.3. Please provide either a CAS verification script or an expanded derivation of the critical inequalities so that the constants can be checked independently.","section":"Section 7, Lemmas 7.2–7.6"}],"minor_comments":[{"comment":"The phrase 'Repeated the process above' would be clearer if the induction invariant and the precise stopping rule for the chain H = H1 ⊊ H2 ⊊ ... ⊊ Hu were stated explicitly.","section":"Section 2, proof of Lemma 2.2"},{"comment":"In the display after (7.9), the polynomial expression is typeset as 'n2(7 − 2t) + 4n(2t2 − 5t − 2) ...'; please insert exponent and multiplication signs for readability.","section":"Section 7, Lemma 7.6(ii)"},{"comment":"The normalization |F| ≤ |G| breaks the symmetry, but the paper does not explicitly say that the extremal description for |F| ≥ |G| is obtained by interchanging F and G; adding this sentence would prevent confusion.","section":"Introduction, Theorem 1.2"},{"comment":"The proposition [11, Proposition 1.6] is invoked repeatedly; stating its content once at the beginning of Section 7 would make the subsequent estimates easier to follow.","section":"Section 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct and the overall structure is coherent. My main concern is the unverified use of [19, Theorem 6] in Lemma 2.4 and the long arithmetic chains in Section 7; both are load-bearing for the main theorem. If the authors add the requested verification, I would be inclined to accept. I also note that several references are by the same research group ([5], [15], [16], [17]); this is not disqualifying, but the novelty relationship with [16] could be clarified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper completes the s-almost cross-t-intersecting characterization for all t, not just t=1. The extremal structures in Theorems 1.2 and 1.3 are new, and the stability results are a useful addition. It's a solid paper.\n\nWhat's genuinely new: [13] gave the maximum product for t=1; [5] handled the cross-t-intersecting case. This paper does the complementary non-cross-t-intersecting case for all t and gives the full structure, including two nontrivial extremal families. The proof method—covering numbers plus a battery of binomial inequalities—is the right tool, and the exposition is clear.\n\nThe soft spots are exactly where the stress-test note points. Lemma 2.4 leans entirely on [19, Theorem 6], an alternating-sequence bound that is quoted but not reproduced, and the hypotheses (distinctness, n-range) are not checked against the constructed sequences. The Section 7 inequalities are long and are stated as verified computations without CAS support. These are the load-bearing dependencies for Theorem 1.1, so a referee should check them. I don't see any internal inconsistency, and the extremal examples do match the upper-bound products.\n\nMinor point: Lemma 2.1 imports [5, Lemma 2.7] from one of the authors' group, but [5] is published, so that's not a real problem. The large-n hypotheses are explicit; yes, they're strong, but that's typical for this style of theorem.\n\nBottom line: this is a serious combinatorics paper that deserves a proper referee, not a desk reject. If you work in extremal set theory, cite it. I'd send it to review, with the instruction that the arithmetic and the [19] bound get a careful look.","headline":"Solid generalization of cross-t-intersecting extremal results for all t, with the main proof depending on an external sequence bound and unverified binomial estimates; worth a careful referee.","tokens_in":24536,"tokens_out":2344,"would_cite":true,"duration_ms":23782,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $k \\geq t+1$ and sufficiently large $n$, every pair of $s$-almost cross-$t$-intersecting families that maximizes $|\\mathcal{F}||\\mathcal{G}|$ has $\\mathcal{F}=\\mathcal{G}$ equal to the full star $\\mathcal{H}_1([n],W;k)$ about a fixed…","keywords":["extremal set theory","cross-t-intersecting families","s-almost cross-t-intersecting","Erdős-Ko-Rado theorem","Hilton-Milner","covering number","product of sizes","stability"],"falsifier":"Exhibit two $s$-almost cross-$t$-intersecting families $\\mathcal{F},\\mathcal{G}\\subseteq\\binom{[n]}{k}$ with $k\\geq t+1$ and $n\\geq (t+1)(2(k-t+1)^2+7s)$ whose product $|\\mathcal{F}||\\mathcal{G}|$ exceeds $\\binom{n-t}{k-t}^2$; that directly refutes Theorem 1.1. Alternatively, check whether two alternating sequences of $k$-sets satisfying $|F_i\\cap G_i|<t$ and $|F_i\\cap G_j|\\geq t$ for $j<i$ can have length greater than $\\binom{2k-2t+2}{k-t+1}$; if so, Lemma 2.4 breaks.","tokens_in":23492,"feed_emoji":"⭐","tokens_out":5678,"duration_ms":54286,"temperature":0.7,"pith_summary":"The paper characterizes the exact pairs of families $\\mathcal{F},\\mathcal{G}\\subseteq \\binom{[n]}{k}$ that maximize the product of their sizes among all $s$-almost cross-$t$-intersecting pairs. For $k\\geq t+1$ and $n\\geq (t+1)(2(k-t+1)^2+7s)$, the unique maximum is attained when both families are the full star of all $k$-sets containing one fixed $t$-set $W$. The paper also describes the extremal structure when the two families are not cross-$t$-intersecting, giving two explicit shapes depending on whether the families live in $\\binom{[n]}{t+1}$ or in larger $k$. A stability corollary then decides which of two competing extremal constructions wins when the common intersection of all sets in both families has size below $t$. The result matters because it settles the product version of the Erdős–Ko–Rado problem under a local, one-sided relaxation of the intersection condition.","feed_headline":"Max product of almost-t-cross-intersecting pairs is a double star","feed_subtitle":"For n large enough, the two families achieving the maximum product are both the full star of k-sets through one fixed t-set.","key_machinery":"The load-bearing object is the $t$-covering number $\\tau_t(\\mathcal{F})$, the minimum size of a set $T$ that meets every member of $\\mathcal{F}$ in at least $t$ points, together with the product bound $f_1(n,k,t,s,x)$ defined in (2.3). Lemma 2.3 bounds $|\\mathcal{F}||\\mathcal{G}|$ by $f_1$ evaluated at the two covering numbers whenever both are at most $k$; Lemma 2.4, using an external bound on alternating sequences of $k$-sets, handles the case where one covering number is at least $k+1$. These bounds force $\\tau_t(\\mathcal{F})=\\tau_t(\\mathcal{G})=t$ under the stated $n$ range, after which Lemma 2.5 shows the unique minimum $t$-covers coincide, giving the double-star conclusion.","core_discovery":"The central claim is Theorem 1.1: if $k\\geq t+1$ and $n\\geq (t+1)(2(k-t+1)^2+7s)$, and $\\mathcal{F},\\mathcal{G}\\subseteq \\binom{[n]}{k}$ are $s$-almost cross-$t$-intersecting with $|\\mathcal{F}||\\mathcal{G}|$ maximum, then there is a $t$-set $W$ such that $\\mathcal{F}=\\mathcal{G}=\\mathcal{H}_1([n],W;k)$, the star of all $k$-sets containing $W$. Theorems 1.2 and 1.3 sharpen this for extremal pairs that fail to be cross-$t$-intersecting: when $k\\geq t+2$, one family is a star with a specific slice of size $\\binom{n-k-1}{k-t}-s$ removed, and the other is the same star with a set of size $\\min\\{t,s\\}$ added; when $k=t+1$, two different shapes arise according to whether $t\\geq s+2$ or $t\\leq s+1$. Combining these with an earlier classification of nearly extremal cross-$t$-intersecting families yields a stability result that determines which extremal shape is optimal when the intersection of all members of both families has size less than $t$.","pith_inferences":["Editorial inference: the same covering-number machinery should extend to a two-parameter version where each family is allowed its own budget, $s_1$ and $s_2$; the extremal shapes would likely be stars with asymmetric removed slices.","Editorial inference: the threshold on $n$ is almost certainly not sharp, because the Section 7 inequalities use deliberately loose constants; a direct comparison of $g_1$, $g_2$, $g_3$, and $g_4$ could identify the true boundary for moderate $n$.","Editorial inference: the method and the extremal shapes should transfer to other ranked posets with a similar intersection notion, such as vector spaces, a direction the concluding remark already points toward.","Editorial inference: a testable quantitative consequence is that for fixed $k,t,s$ the product-maximizing pair should remain a double star for all $n$ beyond some much smaller threshold; searching for the smallest such $n$ by computer would either confirm the trend or reveal a new extremal family."],"forward_implications":["If the theorem is correct, the maximum product for large $n$ is exactly $\\binom{n-t}{k-t}^2$, attained only by $\\mathcal{F}=\\mathcal{G}=\\mathcal{H}_1([n],W;k)$.","Any extremal pair that is not cross-$t$-intersecting must be a star with a removed slice of size $\\binom{n-k-1}{k-t}-s$ together with a small added block, so the extremal families are always close to a full star.","For $k=t+1$, the optimal non-cross-$t$-intersecting shape switches at the boundary $s=t-1$: one family is a singleton for $t\\geq s+2$, and a small star around a $t$-set for $t\\leq s+1$.","The stability corollary determines, under the condition that the common intersection of all sets in both families has size less than $t$, whether the star-with-surgery construction or the cross-intersecting pair $\\mathcal{H}_1([n],Y;k)$ and $\\mathcal{M}_1(Y;k,t)$ wins, depending on whether $k\\geq 2t+1$ or $k\\leq 2t$."],"supporting_citations":[{"why":"This supplies Lemma 2.7 used to bound subfamilies $\\mathcal{I}_H$ and Theorem 1.2 on nearly extremal cross-$t$-intersecting families, which the stability corollaries rely on.","marker":"[5]"},{"why":"This supplies Theorem 6, the external bound $m\\leq \\binom{2k-2t+2}{k-t+1}$ on alternating sequences of $k$-sets, which is the key estimate in Lemma 2.4 for excluding covering numbers at least $k+1$.","marker":"[19]"},{"why":"This supplies Proposition 1.6, used repeatedly in Section 7 to establish the binomial-coefficient inequalities that make the star product dominate all other candidates.","marker":"[11]"},{"why":"This established the $t=1$ case of the maximum-product problem for almost cross-intersecting families, which the present Theorem 1.1 extends to general $t$.","marker":"[13]"}],"fun_headline_variants":["Double star is the extremal shape for s-almost cross-t","Max product of s-almost cross-t: both families are full stars","s-almost cross-t max: double star is the only extremal family pair","Stability result: near-star shapes for s-almost cross-t extremal pairs","Max product forces both families to be the same star on a t-set"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes $n$ is large enough that the full-star product beats every competitor; in particular, the external bound on alternating sequences of $k$-sets from [19] and the Section 7 inequalities must hold in the stated $n$ range, or the contradiction forcing both covering numbers to equal $t$ collapses.","fun_headline_variants_meta":{"raw":{"variants":["Double star is the extremal shape for s-almost cross-t","Max product of s-almost cross-t: both families are full stars","s-almost cross-t max: double star is the only extremal family pair","Stability result: near-star shapes for s-almost cross-t extremal pairs","Max product forces both families to be the same star on a t-set"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001025,"raw_usage":{"total_tokens":4314,"prompt_tokens":933,"completion_tokens":3381,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":3282}},"tokens_in":549,"tokens_out":3381,"duration_ms":27717,"temperature":1.0,"reasoning_tokens":3282,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:16:05.330564+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit two $s$-almost cross-$t$-intersecting families $\\mathcal{F},\\mathcal{G}\\subseteq\\binom{[n]}{k}$ with $k\\geq t+1$ and $n\\geq (t+1)(2(k-t+1)^2+7s)$ whose product $|\\mathcal{F}||\\mathcal{G}|$ exceeds $\\binom{n-t}{k-t}^2$; that directly refutes Theorem 1.1. Alternatively, check whether two alternating sequences of $k$-sets satisfying $|F_i\\cap G_i|<t$ and $|F_i\\cap G_j|\\geq t$ for $j<i$ can have length greater than $\\binom{2k-2t+2}{k-t+1}$; if so, Lemma 2.4 breaks.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This supplies Lemma 2.7 used to bound subfamilies $\\mathcal{I}_H$ and Theorem 1.2 on nearly extremal cross-$t$-intersecting families, which the stability corollaries rely on."},{"cited_title":"Properties of intersecting families of ordered sets","cited_arxiv_id":null,"evidence_quote":"This supplies Theorem 6, the external bound $m\\leq \\binom{2k-2t+2}{k-t+1}$ on alternating sequences of $k$-sets, which is the key estimate in Lemma 2.4 for excluding covering numbers at least $k+1$."},{"cited_title":"Frankl and J","cited_arxiv_id":null,"evidence_quote":"This supplies Proposition 1.6, used repeatedly in Section 7 to establish the binomial-coefficient inequalities that make the star product dominate all other candidates."},{"cited_title":"Gerbner, N","cited_arxiv_id":null,"evidence_quote":"This established the $t=1$ case of the maximum-product problem for almost cross-intersecting families, which the present Theorem 1.1 extends to general $t$."}],"review_version":1}