{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T02:23:56Z","timestamp":1772936636096,"version":"3.50.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T00:00:00Z","timestamp":1772841600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T00:00:00Z","timestamp":1772841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Order"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    A family\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{{\\mathcal {G}}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of sets is a(n induced) copy of a poset\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{P}\\varvec{=}\\varvec{(}\\varvec{P}\\varvec{,}\\varvec{\\leqslant }\\varvec{)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>=<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>,<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u2a7d<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    if there exists a bijection\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{b}\\varvec{:}\\varvec{P}\\varvec{\\rightarrow } \\varvec{{\\mathcal {G}}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>:<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u2192<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>G<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{p}\\varvec{\\leqslant } \\varvec{q}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u2a7d<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    holds if and only if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{b}\\varvec{(p)}\\varvec{\\subset } \\varvec{b}\\varvec{(q)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u2282<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The induced saturation number\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{\\textrm{sat}^*}\\varvec{(}\\varvec{n}\\varvec{,}\\varvec{P}\\varvec{)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:msup>\n                                <mml:mtext>sat<\/mml:mtext>\n                                <mml:mo>\u2217<\/mml:mo>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>,<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is the minimum size of a family\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{{\\mathcal {F}}}\\varvec{\\subseteq } \\varvec{2}^{\\varvec{[n]}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u2286<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    that does not contain any copy of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{P}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , but for any\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{G}\\varvec{\\in } \\varvec{2}^{\\varvec{[n]}}\\varvec{\\setminus } \\varvec{{\\mathcal {F}}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>G<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u2208<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mrow>\n                              <mml:mo>\\<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>F<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the family\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{{\\mathcal {F}}}\\varvec{\\cup } \\varvec{\\{G\\}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u222a<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    contains a copy of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{P}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We consider\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{\\textrm{sat}^*}\\varvec{(n,P)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:msup>\n                                <mml:mtext>sat<\/mml:mtext>\n                                <mml:mo>\u2217<\/mml:mo>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for posets\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{P}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    that are formed by pairwise incomparable chains, i.e.\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{P}\\varvec{=}\\varvec{\\bigoplus }_{\\varvec{j=1}}^{\\varvec{m}}\\varvec{C}_{\\varvec{i}_{\\varvec{j}}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>=<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msubsup>\n                              <mml:mrow>\n                                <mml:mo>\u2a01<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mrow>\n                                  <mml:mi>j<\/mml:mi>\n                                  <mml:mo>=<\/mml:mo>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>m<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi>C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:msub>\n                                <mml:mrow>\n                                  <mml:mi>i<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mrow>\n                                  <mml:mi>j<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We make the following two conjectures: (i)\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{\\textrm{sat}^*}\\varvec{(n,P)}\\varvec{=}\\varvec{O}\\varvec{(n)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:msup>\n                                <mml:mtext>sat<\/mml:mtext>\n                                <mml:mo>\u2217<\/mml:mo>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>=<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>O<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for all such posets and (ii)\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{\\textrm{sat}^*}\\varvec{(n,P)}\\varvec{=}\\varvec{O}\\varvec{(1)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:msup>\n                                <mml:mtext>sat<\/mml:mtext>\n                                <mml:mo>\u2217<\/mml:mo>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>=<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>O<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    if not all two chains are of the same size. (The second conjecture is known to hold if there is a unique longest among the chains.) We verify these conjectures in some special cases: we prove (i) if all chains are of the same length, we prove (ii) in the first unknown general case: for posets\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{2C}_{\\varvec{k}}\\varvec{+}\\varvec{C}_{\\varvec{1}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>+<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi>C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Finally, we give an infinite number of examples showing that (ii) is not a necessary condition for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{\\textrm{sat}^*}\\varvec{(n,P)}\\varvec{=}\\varvec{O}\\varvec{(1)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:msup>\n                                <mml:mtext>sat<\/mml:mtext>\n                                <mml:mo>\u2217<\/mml:mo>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>=<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>O<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    among posets\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{P}\\varvec{=}\\varvec{\\bigoplus }_{\\varvec{j=1}}^{\\varvec{m}}\\varvec{C}_{{\\varvec{i}}_{\\varvec{j}}}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>=<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msubsup>\n                              <mml:mrow>\n                                <mml:mo>\u2a01<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mrow>\n                                  <mml:mi>j<\/mml:mi>\n                                  <mml:mo>=<\/mml:mo>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>m<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi>C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:msub>\n                                <mml:mrow>\n                                  <mml:mi>i<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mrow>\n                                  <mml:mi>j<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    : we prove\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{\\textrm{sat}^*}\\varvec{(}\\varvec{n}\\varvec{,}\\varvec{(}\\left( {\\begin{array}{c}\\varvec{2t}\\\\ \\varvec{t}\\end{array}}\\right) \\varvec{+}\\varvec{1}\\varvec{)}\\varvec{C}_{\\varvec{2}}\\varvec{)}\\varvec{=}\\varvec{O}\\varvec{(1)}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:msup>\n                                <mml:mtext>sat<\/mml:mtext>\n                                <mml:mo>\u2217<\/mml:mo>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>,<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mfenced>\n                              <mml:mrow>\n                                <mml:mtable>\n                                  <mml:mtr>\n                                    <mml:mtd>\n                                      <mml:mrow>\n                                        <mml:mn>2<\/mml:mn>\n                                        <mml:mi>t<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:mtd>\n                                  <\/mml:mtr>\n                                  <mml:mtr>\n                                    <mml:mtd>\n                                      <mml:mrow>\n                                        <mml:mrow\/>\n                                        <mml:mrow>\n                                          <mml:mi>t<\/mml:mi>\n                                        <\/mml:mrow>\n                                      <\/mml:mrow>\n                                    <\/mml:mtd>\n                                  <\/mml:mtr>\n                                <\/mml:mtable>\n                              <\/mml:mrow>\n                            <\/mml:mfenced>\n                            <mml:mrow>\n                              <mml:mo>+<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi>C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>=<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>O<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for all\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{t}\\varvec{\\ge } \\varvec{2}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>\u2265<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    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