{"id":"35935836-6420-4143-9080-a0359a355403","arxiv_id":"1908.07179","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Proper interval graphs are exactly the graphs whose independence complex is sortable, and this implies strong persistence and linear quotients for powers of t-independence ideals of these graphs.","lead":"This paper introduces a new combinatorial property called sortability for simplicial complexes and shows that the independence complex of a graph is sortable exactly when the graph is a proper interval graph. The result yields algebraic consequences: t-independence ideals of proper interval graphs have the strong persistence property and all their powers have linear quotients.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2.6's strong-persistence claim omits the infinite-field hypothesis required by [8, Cor. 1.6]; the rest of the algebraic argument appears sound.","rationale":"I focused on the two central claims: Theorem 1.8 (sortability characterizes proper interval graphs) and Theorem 2.7 (all powers of t-independence ideals of proper interval graphs have linear quotients). The sorting argument in Theorem 1.8 works: sortability forces every closed neighbourhood to be an interval, and the converse uses the clique-interval property correctly. The proof of Theorem 2.7 is also internally sound: the characterization of minimal generators by sorted matrices with independent columns is justified by a pigeonhole argument on the multiset C, and the exchange step for linear quotients correctly replaces the first differing entry by a smaller variable while preserving both the sorted form and column independence. Proposition 2.4's ℓ-exchange proof is consistent, with only a harmless typo in the index range for i. The main weakness I found is the unstated infinite-field hypothesis in Corollary 2.6, which the reader also flagged. This does not undermine the sortability characterization or the linear-quotients theorem, but it does affect the strong-persistence statement as written. Since the reader already returned CONDITIONAL and my concern does not move the verdict, I recommend UNCHANGED.","tokens_in":10637,"tokens_out":41127,"duration_ms":440337,"concrete_test":"Check the exact hypotheses of [8, Corollary 1.6]. If it assumes K infinite, either add 'K infinite' to the statement of Corollary 2.6, or prove the strong-persistence equality I^{k+1}:I = I^k directly over finite K by extending scalars to an infinite field, since monomial ideal containments are field-independent. If the equality does not descend, Corollary 2.6 is false as stated over finite fields; if it does, add the descent remark and the statement becomes fully justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline algebraic package includes the strong persistence property of I_t(G) for proper interval graphs, stated in Corollary 2.6 for arbitrary K. The proof route is: Proposition 2.4 gives the ℓ-exchange property, Theorem 2.5 gives a squarefree Gröbner basis of the Rees ideal, so R(I_t(G)) is normal Cohen-Macaulay, and then [8, Corollary 1.6] yields strong persistence. However, the paper itself notes in the paragraph immediately before Corollary 2.6 that this implication from normality/Cohen-Macaulayness to strong persistence is invoked 'under the assumption that K is infinite'. Corollary 2.6 does not state 'K infinite' in its hypotheses, and no other argument for strong persistence over finite fields is supplied. Since the rest of the paper is otherwise field-agnostic, a reader cannot tell whether the strong-persistence claim is intended for all fields or only infinite ones. This is not an internal inconsistency in the sortability theorem or the linear-quotients proof, both of which I checked and found coherent; it is a genuine gap in the statement of a headline result. If [8, Cor. 1.6] really requires K infinite, the theorem as stated is unsupported for finite K; if the strong-persistence equality I^{k+1}:I = I^k is monomial and field-independent, Corollary 2.6 needs a one-line descent argument to record that.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a notion of sortability for simplicial complexes and proves that the independence complex of a graph is sortable if and only if the graph is a proper interval graph (Theorem 1.8). It then studies the t-independence ideals I_t(G) of proper interval graphs, showing that they satisfy the ℓ-exchange property (Proposition 2.4), that their Rees rings have a quadratic Gröbner basis (Theorem 2.5), and consequently that the ideals have the strong persistence property and all their powers have linear resolutions (Corollary 2.6) and linear quotients (Theorem 2.7). It also proves that the independence complexes of cycle graphs are t-sortable for all t, and that the corresponding toric rings are Koszul and normal Cohen–Macaulay (Corollary 2.2).","tokens_in":10875,"tokens_out":12448,"duration_ms":123601,"significance":"If the results hold, the paper makes a clean contribution: it gives a new combinatorial characterization of proper interval graphs via a sorting operation on faces, and it provides a new family of ideals with strong persistence and linear quotients. The proof of Theorem 1.8 is direct and convincing, and the linear-quotients argument in Theorem 2.7 is essentially self-contained, using the clique-interval property in a nice pigeonhole argument. The algebraic half, however, rests on substantial imported Gröbner-basis results (Theorem 2.1 from Ene–Herzog, and the normality/Cohen–Macaulay implication from [8]), and one of the headline claims, strong persistence, is stated without a field hypothesis that the cited theorem requires. This is a genuine but local gap that a revision can fix.","major_comments":[{"comment":"Corollary 2.6 states that for every proper interval graph G and every t ≥ 2, the ideal I_t(G) satisfies the strong persistence property, with no hypothesis on the field K. The proof invokes [8, Corollary 1.6], but the manuscript itself notes immediately before the corollary that this implication from normality/Cohen–Macaulayness to strong persistence holds 'under the assumption that K is infinite.' Since no alternative argument is provided for finite fields, the strong-persistence claim as stated is unsupported. Please either add 'K infinite' to the hypotheses of Corollary 2.6 or supply a field-descent argument showing that the monomial equality I^{k+1}:I = I^k, which characterizes strong persistence, descends from an infinite extension to K.","section":"Section 2, paragraph before Corollary 2.6 and Corollary 2.6"}],"minor_comments":[{"comment":"The phrase 'Let i_{s,k} be the smallest index such that i_{s,k} ≠ i'_{s,k}' is ambiguous: it should say the first position in the linear order of the sorted entries (equivalently, the lexicographically first pair (s,k)), not the smallest numerical vertex label. The subsequent reasoning is correct once this order is specified.","section":"Theorem 2.7 proof"},{"comment":"In the argument that S is independent, the case s = t is not explicitly addressed. If s = t, then S is exactly the column of u', so the claim follows immediately; spelling this out would improve clarity.","section":"Theorem 2.7 proof"},{"comment":"The corollary calls I_t(G) the 'independence ideal' while the rest of the paper uses 't-independence ideal'; please make the terminology consistent.","section":"Corollary 2.6"},{"comment":"The parity-sensitive interleaving of the two blocks in the join is terse. A short sentence explaining that an odd-length first block shifts the parity of the second block would help the reader verify the formula.","section":"Proposition 1.4 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper overlaps with the authors' earlier work on t-clique ideals, but the sortability characterization of proper interval graphs is a genuinely new angle and the main combinatorial results appear sound. The only load-bearing issue is the omitted infinite-field hypothesis in Corollary 2.6, which should be fixable either by adding the hypothesis or by a short descent argument. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about this paper is that it earns its main claim. The notion of a sortable simplicial complex is new, and Theorem 1.8—independence complex sortable iff proper interval graph—is a real characterization, not present in the cited literature. The proof is short and sound, building on the known graph-theoretic conditions in Lemma 1.7 and a clean sorting argument. That alone is worth the read.\n\nThe algebraic half is mostly solid. The t-independence ideals of proper interval graphs get an ℓ-exchange property (Proposition 2.4) and all powers are shown to have linear quotients (Theorem 2.7). I checked the minimal-generator argument and the exchange step; the pigeonhole contradiction is valid and uses the clique-interval property appropriately. The linear-quotients proof is also coherent. So the paper's central results stand.\n\nThe soft spots are minor but real. Corollary 2.6 states strong persistence for arbitrary fields, but the proof invokes [8, Corollary 1.6], which—as the paper itself notes two paragraphs earlier—requires K infinite. For finite fields, the statement is unsupported as written. This is a genuine gap, not an internal contradiction, and likely fixable: either state the infinite-field hypothesis or give a descent argument for the monomial equality I^{k+1}:I = I^k. The other issue is Remark 1.6's CoCoA computation, which is not reproducible from the paper; it's an aside, so only mildly annoying.\n\nOne thing to keep in mind: Theorem 2.1, the Gröbner-basis result for sorting relations, is imported from Ene–Herzog without proof. That is standard for a paper in this area, but the algebraic results depend on it, so the reader should know the algebraic half rests on that external theorem. The combinatorial half is self-contained.\n\nThis paper is for combinatorial commutative algebraists looking at independence complexes, t-independence ideals, and powers with linear resolutions. It deserves a serious referee—the novel characterization and the linear-quotients results are important enough. The referee should ask for a fix to Corollary 2.6 and a small tightening of the field hypothesis.","headline":"The sortability characterization of proper interval graphs is genuinely new and the proof of linear quotients checks out; the strong-persistence corollary has a small unstated field-hypothesis issue.","tokens_in":11458,"tokens_out":2848,"would_cite":true,"duration_ms":29818,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F20","05E45","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The independence complex of a graph is sortable precisely for proper interval graphs, and this forces strong persistence and linear quotients for t-independence ideals.","keywords":["proper interval graph","sortable simplicial complex","t-sortability","t-independence ideal","strong persistence property","linear quotients","Koszul algebra","monomial ideal"],"falsifier":"Run an exhaustive search over all graphs on, say, six vertices: test whether sorting closure holds for every labeling of the independence complex and compare with the proper interval graph property from Lemma 1.7(v); any graph whose independence complex is sortable but which is not a proper interval graph would refute Theorem 1.8. Equivalently, find a pair of faces in a non-proper interval graph whose sorted pair contains an edge under every labeling.","tokens_in":1666,"feed_emoji":"📐","tokens_out":2083,"duration_ms":68398,"temperature":0.7,"pith_summary":"This paper introduces a combinatorial sorting operation on the faces of a simplicial complex and defines a complex to be sortable when sorting any two faces produces two faces again. Its central result is that the independence complex of a graph is sortable, under some labeling of the vertices, exactly when the graph is a proper interval graph. This graph-theoretic characterization is then used to prove algebraic facts about the t-independence ideal generated by monomials attached to t-vertex independent sets: for proper interval graphs the ideal has the strong persistence property and every power has linear quotients, hence linear resolutions. The interest is that few general families of monomial ideals are known to have strong persistence, and this paper supplies one tied to a recognizable graph class.","feed_headline":"Sortable independence complexes exactly match proper interval graphs","feed_subtitle":"The same sorting condition forces all powers of t-independence ideals to have linear resolutions.","key_machinery":"The sorting operator is the central object: given two faces $F$ and $G$ of a complex, one forms the product monomial $x_F x_G$, writes it with variables in increasing order, and returns the two faces made from the odd-position and even-position variables. A complex is sortable when this operation never leaves the complex, and $t$-sortable when the condition is required only for pairs of faces of size $t$. The algebraic half runs through the sorting relations $y_u y_v - y_{u'} y_{v'}$ attached to unsorted pairs; the paper uses the theorem that, for a sortable monomial ideal, these relations form a Gröbner basis of the defining ideal of the fiber ring with respect to a sorting order, together with the $\\ell$-exchange property, to transfer the combinatorial sorting condition into linear-quotient and persistence statements.","core_discovery":"On the graph-theoretic side, the paper proves (Theorem 1.8) that $\\Delta(G)$ is sortable if and only if $G$ is a proper interval graph, where sorting pairs two faces by writing the multiset union in increasing order and splitting it into odd and even positions. It also proves that every cycle graph is $t$-sortable for every $t$, though cycles are not sortable. On the algebraic side, for any proper interval graph and any $t \\geq 2$, the $t$-independence ideal $I_t(G)$ satisfies the $\\ell$-exchange property, satisfies strong persistence, and each power $I_t(G)^m$ has linear quotients; in addition the fiber ring $K[u : u \\in \\mathcal{G}(I_t(G))]$ is Koszul and a normal Cohen-Macaulay domain. A corollary is that a forest has sortable independence complex exactly when every component is a path.","pith_inferences":["Not pursued in the paper: the sortability criterion might be turned into an algorithmic recognition test for unit interval graphs, since sorting closure can be checked locally on pairs of faces.","The $t$-sortability notion is weaker and applies to cycles, so it may cover further graph classes whose $t$-independence ideals still have Koszul fiber rings; identifying those classes is a natural next step.","Because the strong-persistence conclusion for proper interval graphs passes through normality of the Rees ring, testing whether non-proper interval graphs fail linear quotients for some $t$ would sharpen the boundary of the algebraic theorem."],"forward_implications":["If $G$ is a proper interval graph, then its $t$-independence ideal $I_t(G)$ (for $t \\geq 2$) has the strong persistence property, so $\\operatorname{Ass}(I_t(G)^k) \\subseteq \\operatorname{Ass}(I_t(G)^{k+1})$ for all $k$.","Every power $I_t(G)^m$ has linear quotients with respect to the lex order, and therefore has a linear resolution.","The fiber ring of $I_t(G)$ is Koszul, normal, and Cohen-Macaulay; the same holds for $t$-independence ideals of cycle graphs because cycles are $t$-sortable.","A forest has sortable independence complex exactly when it is a disjoint union of paths; a tree is sortable exactly when it is a path.","The sortability characterization gives a new graph-theoretic recognition principle for proper interval graphs: checking that the sorting operation closes on the independence complex."],"supporting_citations":[{"why":"Supplies the characterization of proper interval graphs used in Lemma 1.7, connecting condition (i) with the proper interval property.","marker":"[11]"},{"why":"Provides the sorting order and the theorem that sorting relations form a Gröbner basis, the bridge from sortability to Koszul and linear-quotient properties.","marker":"[3]"},{"why":"Introduces the $\\ell$-exchange property and the Rees-ring presentation theorem used to conclude linear resolutions for all powers.","marker":"[7]"},{"why":"Gives the criterion that normality or Cohen-Macaulayness of the Rees ring implies strong persistence, used in Corollary 2.6.","marker":"[8]"},{"why":"States the conjecture on powers of $t$-clique ideals that Theorem 2.7 proves for proper interval graphs.","marker":"[10]"},{"why":"Introduces $t$-clique and $t$-independence ideals, defining the objects under study.","marker":"[13]"}],"fun_headline_variants":["Independence complex sortable iff graph is proper interval","Proper interval graphs exactly match sortable independence complexes","Sortability of independence complex pins down proper interval graphs","t-independence ideals of proper interval graphs: all powers have linear quotients","Forests sortable iff paths; cycles t-sortable but never sortable"],"cache_read_input_tokens":13568,"weakest_assumption_plain":"The chain from sortability to strong persistence and linear quotients depends on the imported theorem that, for a sortable monomial ideal, the sorting relations form a Gröbner basis of the defining ideal of the fiber ring under the sorting order; the paper quotes it rather than proving it, and Corollary 2.6 additionally uses a normality-to-strong-persistence criterion that assumes an infinite field.","fun_headline_variants_meta":{"raw":{"variants":["Independence complex sortable iff graph is proper interval","Proper interval graphs exactly match sortable independence complexes","Sortability of independence complex pins down proper interval graphs","t-independence ideals of proper interval graphs: all powers have linear quotients","Forests sortable iff paths; cycles t-sortable but never sortable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001274,"raw_usage":{"total_tokens":5153,"prompt_tokens":832,"completion_tokens":4321,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":4235}},"tokens_in":448,"tokens_out":4321,"duration_ms":31958,"temperature":1.0,"reasoning_tokens":4235,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:23:58.148476+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an exhaustive search over all graphs on, say, six vertices: test whether sorting closure holds for every labeling of the independence complex and compare with the proper interval graph property from Lemma 1.7(v); any graph whose independence complex is sortable but which is not a proper interval graph would refute Theorem 1.8. Equivalently, find a pair of faces in a non-proper interval graph whose sorted pair contains an edge under every labeling.","supporting_citations":[{"cited_title":"Looges and S","cited_arxiv_id":null,"evidence_quote":"Supplies the characterization of proper interval graphs used in Lemma 1.7, connecting condition (i) with the proper interval property."},{"cited_title":"Ene and J","cited_arxiv_id":null,"evidence_quote":"Provides the sorting order and the theorem that sorting relations form a Gröbner basis, the bridge from sortability to Koszul and linear-quotient properties."},{"cited_title":"Herzog, T","cited_arxiv_id":null,"evidence_quote":"Introduces the $\\ell$-exchange property and the Rees-ring presentation theorem used to conclude linear resolutions for all powers."},{"cited_title":"Herzog and A","cited_arxiv_id":null,"evidence_quote":"Gives the criterion that normality or Cohen-Macaulayness of the Rees ring implies strong persistence, used in Corollary 2.6."},{"cited_title":"Khosh-Ahang and S","cited_arxiv_id":null,"evidence_quote":"States the conjecture on powers of $t$-clique ideals that Theorem 2.7 proves for proper interval graphs."},{"cited_title":"Moradi, t-clique ideal and t-independence ideal of a graph , Communications in Algebra 46, (2018), 3377–3387","cited_arxiv_id":null,"evidence_quote":"Introduces $t$-clique and $t$-independence ideals, defining the objects under study."}],"review_version":1}